IODS course project
Maarit Tarkiainen
11/16/2019
About the project
Write a short description about the course and add a link to your GitHub repository here. This is an R Markdown (.Rmd) file so you can use R Markdown syntax.
My expectations
On this course I hope to learn how to use R to produce statistical analysis for my PhD thesis. I hope R could be a tool to produce good figures as well. I have no previous experience in writing code, so I assume I need to work a lot on this course. I hearf of this course from my colleagues.# https://github.com/mjtarkia/IODS-project
<<<<<<< HEAD # Maarit Tarkiainen 11 Nov Linear regression analyses {r child = "chapter2.Rmd"}
install.packages(“ggplot2”) This library is needed to produce good sophisticated pictures
library(ggplot2)This will read the data and name it “learning2014”
learning2014<-read.table("/Users/maarit/IODS-project/data/learning2014")This will show the structure of the data “learning2014”
str(learning2014) ## 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ Age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ Attitude: int 37 31 25 35 37 38 35 29 38 21 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ Points : int 25 12 24 10 22 21 21 31 24 26 ...To see a summery of the variables in “learning2014”
summary (learning2014) ``` To access the GGally and ggplot2 libraries needed in sophisticated plots install.packages(“GGally”)
library(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2library(ggplot2)To explore the relationship of variables to one another, the following codes will produce a plot matrix
pairs (learning2014)
p <- ggpairs(learning2014, mapping = aes(col=gender, alpha=0.3), lower = list(combo = wrap("facethist", bins = 20)))
p
This picture shows correlations of all variables to each other, one by one. Both genders have been presented separately. #Age has the most non-normal distribution, with skewness to right.Attitude, surf and strategic questions show biggest difference between genders.
my_model <- lm(Points ~ Attitude + stra + surf, data = learning2014)
summary(my_model)##
## Call:
## lm(formula = Points ~ Attitude + stra + surf, data = learning2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.1550 -3.4346 0.5156 3.6401 10.8952
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.01711 3.68375 2.991 0.00322 **
## Attitude 0.33952 0.05741 5.913 1.93e-08 ***
## stra 0.85313 0.54159 1.575 0.11716
## surf -0.58607 0.80138 -0.731 0.46563
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.296 on 162 degrees of freedom
## Multiple R-squared: 0.2074, Adjusted R-squared: 0.1927
## F-statistic: 14.13 on 3 and 162 DF, p-value: 3.156e-08The only factor in this model that has a significant p-value is Attitude. Thus, the exploration continues.
my_model2 <- lm(Points ~ Attitude + Age + deep, data = learning2014)
summary(my_model2)##
## Call:
## lm(formula = Points ~ Attitude + Age + deep, data = learning2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.206 -3.430 0.204 3.979 10.950
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.60773 3.38966 4.605 8.32e-06 ***
## Attitude 0.35941 0.05696 6.310 2.56e-09 ***
## Age -0.07716 0.05322 -1.450 0.149
## deep -0.60275 0.75031 -0.803 0.423
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.307 on 162 degrees of freedom
## Multiple R-squared: 0.2043, Adjusted R-squared: 0.1896
## F-statistic: 13.87 on 3 and 162 DF, p-value: 4.305e-08Also in this model, only Attitude is significant
my_model5 <- lm(Points ~ Attitude + stra, data = learning2014)
summary(my_model5)##
## Call:
## lm(formula = Points ~ Attitude + stra, data = learning2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.6436 -3.3113 0.5575 3.7928 10.9295
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.97290 2.39591 3.745 0.00025 ***
## Attitude 0.34658 0.05652 6.132 6.31e-09 ***
## stra 0.91365 0.53447 1.709 0.08927 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.289 on 163 degrees of freedom
## Multiple R-squared: 0.2048, Adjusted R-squared: 0.1951
## F-statistic: 20.99 on 2 and 163 DF, p-value: 7.734e-09The adjusted R square of this model is 0.1951, meaning that 19.5% of the change in points is explained by the variables in the model. It means that the model is not very good in explaning the variation in the test variable points. Coefficient estimate 0.34658 shows the change in Points when Attitude changes 1 unit. Let’s allow R print 4 different plot on same page And print residuals vs fitted, Normal QQ-plot, and Residuals vs Leverage to test the assumptions of the model:
par(mfrow = c(2, 2), oma = c(0, 0, 2, 0))
plot(my_model5, which = c(1,2,5)) 
The plots show that the residuals of the model are close to normally distributed.
Week 3 Logistic regression
First let’s rea the data file that we’re working on:
alc<-read.table("/Users/maarit/IODS-project/data/create_alc", sep=" ", header=TRUE)
str(alc)## 'data.frame': 382 obs. of 35 variables:
## $ school : Factor w/ 2 levels "GP","MS": 1 1 1 1 1 1 1 1 1 1 ...
## $ sex : Factor w/ 2 levels "F","M": 1 1 1 1 1 2 2 1 2 2 ...
## $ age : int 18 17 15 15 16 16 16 17 15 15 ...
## $ address : Factor w/ 2 levels "R","U": 2 2 2 2 2 2 2 2 2 2 ...
## $ famsize : Factor w/ 2 levels "GT3","LE3": 1 1 2 1 1 2 2 1 2 1 ...
## $ Pstatus : Factor w/ 2 levels "A","T": 1 2 2 2 2 2 2 1 1 2 ...
## $ Medu : int 4 1 1 4 3 4 2 4 3 3 ...
## $ Fedu : int 4 1 1 2 3 3 2 4 2 4 ...
## $ Mjob : Factor w/ 5 levels "at_home","health",..: 1 1 1 2 3 4 3 3 4 3 ...
## $ Fjob : Factor w/ 5 levels "at_home","health",..: 5 3 3 4 3 3 3 5 3 3 ...
## $ reason : Factor w/ 4 levels "course","home",..: 1 1 3 2 2 4 2 2 2 2 ...
## $ nursery : Factor w/ 2 levels "no","yes": 2 1 2 2 2 2 2 2 2 2 ...
## $ internet : Factor w/ 2 levels "no","yes": 1 2 2 2 1 2 2 1 2 2 ...
## $ guardian : Factor w/ 3 levels "father","mother",..: 2 1 2 2 1 2 2 2 2 2 ...
## $ traveltime: int 2 1 1 1 1 1 1 2 1 1 ...
## $ studytime : int 2 2 2 3 2 2 2 2 2 2 ...
## $ failures : int 0 0 2 0 0 0 0 0 0 0 ...
## $ schoolsup : Factor w/ 2 levels "no","yes": 2 1 2 1 1 1 1 2 1 1 ...
## $ famsup : Factor w/ 2 levels "no","yes": 1 2 1 2 2 2 1 2 2 2 ...
## $ paid : Factor w/ 2 levels "no","yes": 1 1 2 2 2 2 1 1 2 2 ...
## $ activities: Factor w/ 2 levels "no","yes": 1 1 1 2 1 2 1 1 1 2 ...
## $ higher : Factor w/ 2 levels "no","yes": 2 2 2 2 2 2 2 2 2 2 ...
## $ romantic : Factor w/ 2 levels "no","yes": 1 1 1 2 1 1 1 1 1 1 ...
## $ famrel : int 4 5 4 3 4 5 4 4 4 5 ...
## $ freetime : int 3 3 3 2 3 4 4 1 2 5 ...
## $ goout : int 4 3 2 2 2 2 4 4 2 1 ...
## $ Dalc : int 1 1 2 1 1 1 1 1 1 1 ...
## $ Walc : int 1 1 3 1 2 2 1 1 1 1 ...
## $ health : int 3 3 3 5 5 5 3 1 1 5 ...
## $ absences : int 5 3 8 1 2 8 0 4 0 0 ...
## $ G1 : int 2 7 10 14 8 14 12 8 16 13 ...
## $ G2 : int 8 8 10 14 12 14 12 9 17 14 ...
## $ G3 : int 8 8 11 14 12 14 12 10 18 14 ...
## $ alc_use : num 1 1 2.5 1 1.5 1.5 1 1 1 1 ...
## $ high_use : logi FALSE FALSE TRUE FALSE FALSE FALSE ...The data file shows students’ conditions, such as their parents’ education status, the time they spent on studying, travelling. Their social status, absencesetc. Alcohol consumption has been evaluated, and denominated to a high alcohol use group.
To induce a plot which describes high alcohol use, first access some libraries in R
library(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:GGally':
##
## nasa## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(ggplot2)To examine the relation of different variables, such as age, school absence, final grades, or quality of family relationships with high alcohol consumption, boxplot figures for each of the were produced. Sex was used as a grouping factor. The hypothesis was that young age, high school absence, low final grades and low quality of family relationships would correlate with high alcohol consumption.
g1 <- ggplot(alc, aes(x = high_use, y = age, col=sex))
g1 + geom_boxplot()
alc %>% group_by(sex, high_use) %>% summarise(count = n())## # A tibble: 4 x 3
## # Groups: sex [2]
## sex high_use count
## <fct> <lgl> <int>
## 1 F FALSE 156
## 2 F TRUE 42
## 3 M FALSE 112
## 4 M TRUE 72It seems that high alcohol use was more present in males. Males consuming more were older than low consuming males, whereas in females, the older consumed less alcohol
g2 <- ggplot(alc, aes(x = high_use, y = absences))
g2 + geom_boxplot()
It seems that students in high use group have more absence from school than students than other students.
alc %>% group_by(high_use) %>% summarise(count = n(), School_absence=mean(absences))## # A tibble: 2 x 3
## high_use count School_absence
## <lgl> <int> <dbl>
## 1 FALSE 268 3.71
## 2 TRUE 114 6.37g3 <- ggplot(alc, aes(x = high_use, y = G3))
g3 + geom_boxplot() + ylab("grade")
alc %>% group_by(high_use) %>% summarise(count = n(), mean_grade = mean(G3))## # A tibble: 2 x 3
## high_use count mean_grade
## <lgl> <int> <dbl>
## 1 FALSE 268 11.7
## 2 TRUE 114 10.8Students in
g4 <- ggplot(alc, aes(x = high_use, y = famrel))
g4 + geom_boxplot() + ylab("Quality of family relationships")
alc %>% group_by(high_use) %>% summarise(count = n(), Family_relationships = mean(famrel))## # A tibble: 2 x 3
## high_use count Family_relationships
## <lgl> <int> <dbl>
## 1 FALSE 268 4.00
## 2 TRUE 114 3.78Forming a logistic model which includes all variables age, sex, absences, grades, family relationships, and print it
m <- glm(high_use ~ age + absences + famrel+G3, data = alc, family = "binomial")
print(m)##
## Call: glm(formula = high_use ~ age + absences + famrel + G3, family = "binomial",
## data = alc)
##
## Coefficients:
## (Intercept) age absences famrel G3
## -2.06975 0.14944 0.07981 -0.24448 -0.06241
##
## Degrees of Freedom: 381 Total (i.e. Null); 377 Residual
## Null Deviance: 465.7
## Residual Deviance: 437.2 AIC: 447.2To see the coefficients of factor type variables directly, without an intercept, I have fitter the model by adding - 1 to the model formula. Following also coefficients of the model
m1 <- glm(high_use ~ age + absences + sex-1 + famrel + G3, data = alc, family = "binomial")
coef(m1)## age absences sexF sexM famrel G3
## 0.15270276 0.08669679 -2.53225653 -1.49900250 -0.29051313 -0.06176356To calculate odds ratios and their confidence intervals, and print them out
OR <- coef(m1) %>% exp
CI<-confint(m1)%>%exp## Waiting for profiling to be done...cbind(OR, CI)## OR 2.5 % 97.5 %
## age 1.16497865 0.95043571 1.4323988
## absences 1.09056596 1.04507712 1.1426466
## sexF 0.07947947 0.00185329 3.2525841
## sexM 0.22335284 0.00528570 9.2331481
## famrel 0.74787971 0.58049882 0.9616909
## G3 0.94010513 0.87449700 1.0095533The 95% confidence intervals of family relationships and school absences don’t include zero. Thus, only the primary hypotheses that higher amount of school absences and poorer family relationships predict higher alcohol consumption.
To predict high_use, let’s first build a model with only the two significant factors absences and family relationships
m7 <- glm(high_use ~ absences + famrel, data = alc, family = "binomial")
print(m7)##
## Call: glm(formula = high_use ~ absences + famrel, family = "binomial",
## data = alc)
##
## Coefficients:
## (Intercept) absences famrel
## -0.33027 0.08668 -0.24109
##
## Degrees of Freedom: 381 Total (i.e. Null); 379 Residual
## Null Deviance: 465.7
## Residual Deviance: 443.7 AIC: 449.7Then we will predict the probability of high alcohol use, and add the predictive variable into the data. FInally, see the last ten classes and predicted probabilities, as well as class prediction:
probabilities <- predict(m7, type = "response")
alc <- mutate(alc, probability = probabilities)
alc <- mutate(alc, prediction = probability > 0.5)
select(alc, absences, high_use, famrel, probability, prediction) %>% tail(10)## absences high_use famrel probability prediction
## 373 0 FALSE 4 0.2150733 FALSE
## 374 7 TRUE 5 0.2831342 FALSE
## 375 1 FALSE 5 0.1901522 FALSE
## 376 6 FALSE 4 0.3154941 FALSE
## 377 2 FALSE 5 0.2038593 FALSE
## 378 2 FALSE 4 0.2457776 FALSE
## 379 2 FALSE 2 0.3454525 FALSE
## 380 3 FALSE 1 0.4227907 FALSE
## 381 4 TRUE 2 0.3856256 FALSE
## 382 2 TRUE 4 0.2457776 FALSETable regarding actual vs predicted values:
table(high_use = alc$high_use, prediction = alc$prediction)## prediction
## high_use FALSE TRUE
## FALSE 259 9
## TRUE 101 13Let’s draw a picture showing the actual and predicted values
g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
g + geom_point()
Let us estimate the average number of wrong predictions (training error):
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.2879581##Bonus:A ten-fold cross-validation of the model, using functions in boot-library:
library(boot)
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m7, K = 10)
cv$delta[1]## [1] 0.2905759The error is 0.29, which is larger than the 0.26 error in DataCamp exercises. Thus, this model is not as good as in the examples. The smaller number of variables in this model might contribute to the larger error.
##Super-bonus: comparing the prediction error of a model with several variables to a model with less variables. Let’s choose the following variables to the first model: age, sex, no of past class failures, absences, famrel, studytime, final grades (G3).
m2 <- glm(high_use ~ age + sex + failures + absences + famrel + studytime+ G3, data = alc, family = "binomial")
coef(m2)## (Intercept) age sexM failures absences famrel
## -1.99651236 0.14192390 0.86573372 0.21947105 0.08060070 -0.27823424
## studytime G3
## -0.32641486 -0.03484995Testing the model against predicted
probabilities <- predict(m2, type = "response")
alc <- mutate(alc, probability = probabilities)
alc <- mutate(alc, prediction = probability > 0.5)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.2591623The training error of model m2 is 0.251. The following will show the testing error for m2:
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m2, K = 10)
cv$delta[1]## [1] 0.2670157Testing error is a little higher, 0.259
Next let us create a model (m3)with less variables: age, sex, no of past class failures, absences, final grades (G3).
m3 <- glm(high_use ~ age + sex + failures + absences + G3, data = alc, family = "binomial")
coef(m3)## (Intercept) age sexM failures absences G3
## -3.36159261 0.12110210 0.95711056 0.31893668 0.08900606 -0.04528170The training errors of model m3:
probabilities <- predict(m3, type = "response")
alc <- mutate(alc, probability = probabilities)
alc <- mutate(alc, prediction = probability > 0.5)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.2513089Testing error for m3:
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m3, K = 10)
cv$delta[1]## [1] 0.2539267Here, the training error is similar to the model with more variables. However, the testing error seems to increase when the number of variables decreases. With only two variables in model m7 (see above)
Clustering analysis
1## Item 2 To start working, let’s get an access to the Boston file in MASS library, and explore its structure and function:
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectdata("Boston")
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14The data includes 506 observations of 14 variables regarding the housing values of suburbs in Boston. The variables include nitrogen oxides consentration, crime rate per capita, ratio of teachers/student, household sizes etc.
Next, let’s have a summary of the data
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Item 3
Here we’ll have a plot on the data Boston
pairs(Boston)
Scaling this plot larger one could see that medv has a negatie correlation with lstat, a positive correlation with rm. Also a negative correlation for lstat and rm, as well as for dis and age.
cor_matrix<-cor(Boston)
print(cor_matrix)## crim zn indus chas nox
## crim 1.00000000 -0.20046922 0.40658341 -0.055891582 0.42097171
## zn -0.20046922 1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus 0.40658341 -0.53382819 1.00000000 0.062938027 0.76365145
## chas -0.05589158 -0.04269672 0.06293803 1.000000000 0.09120281
## nox 0.42097171 -0.51660371 0.76365145 0.091202807 1.00000000
## rm -0.21924670 0.31199059 -0.39167585 0.091251225 -0.30218819
## age 0.35273425 -0.56953734 0.64477851 0.086517774 0.73147010
## dis -0.37967009 0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad 0.62550515 -0.31194783 0.59512927 -0.007368241 0.61144056
## tax 0.58276431 -0.31456332 0.72076018 -0.035586518 0.66802320
## ptratio 0.28994558 -0.39167855 0.38324756 -0.121515174 0.18893268
## black -0.38506394 0.17552032 -0.35697654 0.048788485 -0.38005064
## lstat 0.45562148 -0.41299457 0.60379972 -0.053929298 0.59087892
## medv -0.38830461 0.36044534 -0.48372516 0.175260177 -0.42732077
## rm age dis rad tax
## crim -0.21924670 0.35273425 -0.37967009 0.625505145 0.58276431
## zn 0.31199059 -0.56953734 0.66440822 -0.311947826 -0.31456332
## indus -0.39167585 0.64477851 -0.70802699 0.595129275 0.72076018
## chas 0.09125123 0.08651777 -0.09917578 -0.007368241 -0.03558652
## nox -0.30218819 0.73147010 -0.76923011 0.611440563 0.66802320
## rm 1.00000000 -0.24026493 0.20524621 -0.209846668 -0.29204783
## age -0.24026493 1.00000000 -0.74788054 0.456022452 0.50645559
## dis 0.20524621 -0.74788054 1.00000000 -0.494587930 -0.53443158
## rad -0.20984667 0.45602245 -0.49458793 1.000000000 0.91022819
## tax -0.29204783 0.50645559 -0.53443158 0.910228189 1.00000000
## ptratio -0.35550149 0.26151501 -0.23247054 0.464741179 0.46085304
## black 0.12806864 -0.27353398 0.29151167 -0.444412816 -0.44180801
## lstat -0.61380827 0.60233853 -0.49699583 0.488676335 0.54399341
## medv 0.69535995 -0.37695457 0.24992873 -0.381626231 -0.46853593
## ptratio black lstat medv
## crim 0.2899456 -0.38506394 0.4556215 -0.3883046
## zn -0.3916785 0.17552032 -0.4129946 0.3604453
## indus 0.3832476 -0.35697654 0.6037997 -0.4837252
## chas -0.1215152 0.04878848 -0.0539293 0.1752602
## nox 0.1889327 -0.38005064 0.5908789 -0.4273208
## rm -0.3555015 0.12806864 -0.6138083 0.6953599
## age 0.2615150 -0.27353398 0.6023385 -0.3769546
## dis -0.2324705 0.29151167 -0.4969958 0.2499287
## rad 0.4647412 -0.44441282 0.4886763 -0.3816262
## tax 0.4608530 -0.44180801 0.5439934 -0.4685359
## ptratio 1.0000000 -0.17738330 0.3740443 -0.5077867
## black -0.1773833 1.00000000 -0.3660869 0.3334608
## lstat 0.3740443 -0.36608690 1.0000000 -0.7376627
## medv -0.5077867 0.33346082 -0.7376627 1.0000000Let us have a correlation matrix plot
library(corrplot)## corrplot 0.84 loadedcorrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex=0.6)
In this plot, blue means a positive correlation, as for rad and tax. Red means negative correlation, as for lstat/medv and dis with indux, nox or age.
Item 4
Next we will standardize the Boston data.
Boston_scaled<-scale(Boston)
summary(Boston_scaled)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865To create crime as categorical variable, first let’s make a vector “rob” to separate based on crime level
Boston_scaled<-as.data.frame(Boston_scaled)
class(Boston_scaled)## [1] "data.frame"rob<-quantile(Boston_scaled$crim)To create the categorical vector “crime” with (“low”, “med_low”, “med_high”, “high”) as cut points, I’ll use cut command:
crime <- cut(Boston_scaled$crim, breaks = rob, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))
table(crime)## crime
## low med_low med_high high
## 127 126 126 127Now let’s remove original “crim” variable from the scaled dataset and add the new “crime” instead
Boston_scaled <- dplyr::select(Boston_scaled, -crim)
Boston_scaled <- data.frame(Boston_scaled, crime)To form test and train sets, first get rownumbers for data set, and take a sample of 80 % of rows. Then create a train set, which includes the selected rownumbers, and a test set which does not include them.
n <- nrow(Boston_scaled)
ind <- sample(n, size = n * 0.8)
train <- Boston_scaled[ind,]
test <- Boston_scaled[-ind,]Item 5
Let’s explore a linear discriminant analysis on “crime”
lda.fit <- lda(crime ~ ., data = train)
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2623762 0.2400990 0.2400990 0.2574257
##
## Group means:
## zn indus chas nox rm
## low 1.07595504 -0.9574546 -0.12375925 -0.9099443 0.4625262
## med_low -0.05360647 -0.2748461 -0.06938576 -0.5892653 -0.0806049
## med_high -0.37054365 0.1871541 0.25532354 0.4311852 0.1849266
## high -0.48724019 1.0149946 -0.04518867 1.0292447 -0.3999330
## age dis rad tax ptratio
## low -0.9173063 0.9439910 -0.6947544 -0.7386615 -0.50106521
## med_low -0.3558275 0.3849555 -0.5532680 -0.4896596 -0.04374653
## med_high 0.4975516 -0.3973782 -0.3934299 -0.3118408 -0.48231855
## high 0.8017165 -0.8420818 1.6596029 1.5294129 0.80577843
## black lstat medv
## low 0.3767155 -0.7910597925 0.53228663
## med_low 0.3206489 -0.1805546555 0.01089293
## med_high 0.1038867 0.0005238553 0.25032284
## high -0.7729866 0.8350022100 -0.62520919
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.126366057 0.6513681475 -0.88511407
## indus -0.003372489 -0.3494731650 0.45997086
## chas -0.069367420 -0.0900761019 -0.03512166
## nox 0.302479237 -0.6392832734 -1.20270539
## rm -0.103959732 -0.1081008439 -0.10801880
## age 0.303704999 -0.4561585337 -0.22716158
## dis -0.142003274 -0.3222008255 0.16833934
## rad 3.396492665 0.7578984946 -0.19543523
## tax 0.078852558 0.1123845467 0.43033494
## ptratio 0.137082492 0.2073145559 -0.04824372
## black -0.156308077 -0.0005323527 0.13183434
## lstat 0.163581193 -0.1380820145 0.37911922
## medv 0.183777413 -0.2679892224 -0.09024372
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9518 0.0372 0.0110A biplot of the LDA analysis
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "orange", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}To set the variable crime as numeric and draw a plot
classes <- as.numeric(train$crime)
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)
### Item 6
classes1 <- as.numeric(test$crime)
test<-data.frame(test, classes1)
test<-dplyr::select (test, -crime)Item 7
Reload the data “Boston”
library(MASS)
data("Boston")
scale(Boston)## crim zn indus chas nox
## 1 -0.419366929 0.28454827 -1.28663623 -0.2723291 -0.14407485
## 2 -0.416926670 -0.48724019 -0.59279438 -0.2723291 -0.73953036
## 3 -0.416928995 -0.48724019 -0.59279438 -0.2723291 -0.73953036
## 4 -0.416338404 -0.48724019 -1.30558569 -0.2723291 -0.83445805
## 5 -0.412074053 -0.48724019 -1.30558569 -0.2723291 -0.83445805
## 6 -0.416631374 -0.48724019 -1.30558569 -0.2723291 -0.83445805
## 7 -0.409837246 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 8 -0.403296561 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 9 -0.395543302 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 10 -0.400333140 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 11 -0.393956378 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 12 -0.406444832 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 13 -0.409198989 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 14 -0.346886928 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 15 -0.345933611 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 16 -0.347162460 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 17 -0.297573695 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 18 -0.328932014 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 19 -0.326780075 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 20 -0.335721492 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 21 -0.274570851 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 22 -0.321045059 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 23 -0.276816959 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 24 -0.305188606 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 25 -0.332877817 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 26 -0.322382028 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 27 -0.341986646 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 28 -0.308985598 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 29 -0.330235268 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 30 -0.303558666 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 31 -0.288635766 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 32 -0.262604396 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 33 -0.258736486 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 34 -0.286204807 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 35 -0.232598159 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 36 -0.412641393 -0.48724019 -0.75459363 -0.2723291 -0.48063666
## 37 -0.408773484 -0.48724019 -0.75459363 -0.2723291 -0.48063666
## 38 -0.410784750 -0.48724019 -0.75459363 -0.2723291 -0.48063666
## 39 -0.399750686 -0.48724019 -0.75459363 -0.2723291 -0.48063666
## 40 -0.416889467 2.72854505 -1.19334657 -0.2723291 -1.09335175
## 41 -0.416196569 2.72854505 -1.19334657 -0.2723291 -1.09335175
## 42 -0.405285738 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 43 -0.403651148 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 44 -0.401574777 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 45 -0.405837965 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 46 -0.400172703 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 47 -0.398203290 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 48 -0.393447167 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 49 -0.390587216 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 50 -0.394551620 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 51 -0.409786092 0.41317968 -0.80123846 -0.2723291 -0.99842406
## 52 -0.415059564 0.41317968 -0.80123846 -0.2723291 -0.99842406
## 53 -0.413870242 0.41317968 -0.80123846 -0.2723291 -0.99842406
## 54 -0.414310861 0.41317968 -0.80123846 -0.2723291 -0.99842406
## 55 -0.418520570 2.72854505 -1.04029322 -0.2723291 -1.24868797
## 56 -0.418577536 3.37170210 -1.44552019 -0.2723291 -1.30909650
## 57 -0.417712575 3.15731641 -1.51548743 -0.2723291 -1.24868797
## 58 -0.418436864 3.80047346 -1.43094368 -0.2723291 -1.24005818
## 59 -0.402145605 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 60 -0.408094536 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 61 -0.402742009 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 62 -0.400138988 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 63 -0.407281891 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 64 -0.405395021 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 65 -0.417833484 0.26310970 -1.42219777 -0.2723291 -1.19604625
## 66 -0.415934988 2.94293073 -1.13212523 -0.2723291 -1.35224545
## 67 -0.415010735 2.94293073 -1.13212523 -0.2723291 -1.35224545
## 68 -0.413371495 0.04872402 -0.73855947 -0.2723291 -1.25731776
## 69 -0.404344047 0.04872402 -0.73855947 -0.2723291 -1.25731776
## 70 -0.405202032 0.04872402 -0.73855947 -0.2723291 -1.25731776
## 71 -0.409840734 -0.48724019 -0.04763292 -0.2723291 -1.22279860
## 72 -0.401644532 -0.48724019 -0.04763292 -0.2723291 -1.22279860
## 73 -0.409447781 -0.48724019 -0.04763292 -0.2723291 -1.22279860
## 74 -0.397385995 -0.48724019 -0.04763292 -0.2723291 -1.22279860
## 75 -0.410921935 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 76 -0.409043203 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 77 -0.408297988 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 78 -0.409979081 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 79 -0.413537744 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 80 -0.410351107 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 81 -0.415319982 0.58468822 -0.91493524 -0.2723291 -1.11061133
## 82 -0.414914241 0.58468822 -0.91493524 -0.2723291 -1.11061133
## 83 -0.415847794 0.58468822 -0.91493524 -0.2723291 -1.11061133
## 84 -0.415973353 0.58468822 -0.91493524 -0.2723291 -1.11061133
## 85 -0.414220179 -0.48724019 -0.96886832 -0.2723291 -0.91212616
## 86 -0.413434274 -0.48724019 -0.96886832 -0.2723291 -0.91212616
## 87 -0.414070206 -0.48724019 -0.96886832 -0.2723291 -0.91212616
## 88 -0.411788058 -0.48724019 -0.96886832 -0.2723291 -0.91212616
## 89 -0.413521468 -0.48724019 -1.12629463 -0.2723291 -0.56693456
## 90 -0.413937672 -0.48724019 -1.12629463 -0.2723291 -0.56693456
## 91 -0.414656148 -0.48724019 -1.12629463 -0.2723291 -0.56693456
## 92 -0.415530409 -0.48724019 -1.12629463 -0.2723291 -0.56693456
## 93 -0.415215350 0.71331963 0.56895343 -0.2723291 -0.78267931
## 94 -0.416759258 0.71331963 0.56895343 -0.2723291 -0.78267931
## 95 -0.415109555 0.71331963 0.56895343 -0.2723291 -0.78267931
## 96 -0.405913532 -0.48724019 -1.20209248 -0.2723291 -0.94664532
## 97 -0.406727340 -0.48724019 -1.20209248 -0.2723291 -0.94664532
## 98 -0.406054205 -0.48724019 -1.20209248 -0.2723291 -0.94664532
## 99 -0.410583624 -0.48724019 -1.20209248 -0.2723291 -0.94664532
## 100 -0.412126370 -0.48724019 -1.20209248 -0.2723291 -0.94664532
## 101 -0.402818740 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 102 -0.406811046 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 103 -0.393506459 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 104 -0.395500287 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 105 -0.403872039 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 106 -0.404683521 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 107 -0.400198280 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 108 -0.404852095 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 109 -0.405218308 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 110 -0.389452536 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 111 -0.407553935 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 112 -0.408378206 -0.48724019 -0.16424500 -0.2723291 -0.06640675
## 113 -0.405768210 -0.48724019 -0.16424500 -0.2723291 -0.06640675
## 114 -0.394278413 -0.48724019 -0.16424500 -0.2723291 -0.06640675
## 115 -0.403556979 -0.48724019 -0.16424500 -0.2723291 -0.06640675
## 116 -0.400182004 -0.48724019 -0.16424500 -0.2723291 -0.06640675
## 117 -0.404804429 -0.48724019 -0.16424500 -0.2723291 -0.06640675
## 118 -0.402549021 -0.48724019 -0.16424500 -0.2723291 -0.06640675
## 119 -0.404920687 -0.48724019 -0.16424500 -0.2723291 -0.06640675
## 120 -0.403272146 -0.48724019 -0.16424500 -0.2723291 -0.06640675
## 121 -0.412081029 -0.48724019 2.11552109 -0.2723291 0.22700611
## 122 -0.411771782 -0.48724019 2.11552109 -0.2723291 0.22700611
## 123 -0.409290833 -0.48724019 2.11552109 -0.2723291 0.22700611
## 124 -0.402618775 -0.48724019 2.11552109 -0.2723291 0.22700611
## 125 -0.408651413 -0.48724019 2.11552109 -0.2723291 0.22700611
## 126 -0.400451723 -0.48724019 2.11552109 -0.2723291 0.22700611
## 127 -0.375069074 -0.48724019 2.11552109 -0.2723291 0.22700611
## 128 -0.389973373 -0.48724019 1.56744433 -0.2723291 0.59808708
## 129 -0.382267781 -0.48724019 1.56744433 -0.2723291 0.59808708
## 130 -0.317649158 -0.48724019 1.56744433 -0.2723291 0.59808708
## 131 -0.380566923 -0.48724019 1.56744433 -0.2723291 0.59808708
## 132 -0.281412645 -0.48724019 1.56744433 -0.2723291 0.59808708
## 133 -0.351503540 -0.48724019 1.56744433 -0.2723291 0.59808708
## 134 -0.381757407 -0.48724019 1.56744433 -0.2723291 0.59808708
## 135 -0.306613931 -0.48724019 1.56744433 -0.2723291 0.59808708
## 136 -0.355255192 -0.48724019 1.56744433 -0.2723291 0.59808708
## 137 -0.382592141 -0.48724019 1.56744433 -0.2723291 0.59808708
## 138 -0.379140436 -0.48724019 1.56744433 -0.2723291 0.59808708
## 139 -0.391060387 -0.48724019 1.56744433 -0.2723291 0.59808708
## 140 -0.356796775 -0.48724019 1.56744433 -0.2723291 0.59808708
## 141 -0.386282176 -0.48724019 1.56744433 -0.2723291 0.59808708
## 142 -0.230758955 -0.48724019 1.56744433 -0.2723291 0.59808708
## 143 -0.034002444 -0.48724019 1.23072696 3.6647712 2.72964520
## 144 0.056254596 -0.48724019 1.23072696 -0.2723291 2.72964520
## 145 -0.096934161 -0.48724019 1.23072696 -0.2723291 2.72964520
## 146 -0.143483937 -0.48724019 1.23072696 -0.2723291 2.72964520
## 147 -0.169559485 -0.48724019 1.23072696 -0.2723291 2.72964520
## 148 -0.144730225 -0.48724019 1.23072696 -0.2723291 2.72964520
## 149 -0.149105020 -0.48724019 1.23072696 -0.2723291 2.72964520
## 150 -0.102255298 -0.48724019 1.23072696 -0.2723291 2.72964520
## 151 -0.227508376 -0.48724019 1.23072696 -0.2723291 2.72964520
## 152 -0.246142237 -0.48724019 1.23072696 -0.2723291 2.72964520
## 153 -0.289127538 -0.48724019 1.23072696 3.6647712 2.72964520
## 154 -0.170241920 -0.48724019 1.23072696 -0.2723291 2.72964520
## 155 -0.255730050 -0.48724019 1.23072696 3.6647712 2.72964520
## 156 -0.009127843 -0.48724019 1.23072696 3.6647712 2.72964520
## 157 -0.135655111 -0.48724019 1.23072696 -0.2723291 2.72964520
## 158 -0.277850494 -0.48724019 1.23072696 -0.2723291 0.43412107
## 159 -0.263985543 -0.48724019 1.23072696 -0.2723291 0.43412107
## 160 -0.254431446 -0.48724019 1.23072696 -0.2723291 2.72964520
## 161 -0.272051536 -0.48724019 1.23072696 3.6647712 0.43412107
## 162 -0.249974107 -0.48724019 1.23072696 -0.2723291 0.43412107
## 163 -0.206910914 -0.48724019 1.23072696 3.6647712 0.43412107
## 164 -0.243503177 -0.48724019 1.23072696 3.6647712 0.43412107
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## 399 4.038608880 -0.48724019 1.01499462 -0.2723291 1.19354259
## 400 0.732778398 -0.48724019 1.01499462 -0.2723291 1.19354259
## 401 2.491712382 -0.48724019 1.01499462 -0.2723291 1.19354259
## 402 1.234973056 -0.48724019 1.01499462 -0.2723291 1.19354259
## 403 0.695478123 -0.48724019 1.01499462 -0.2723291 1.19354259
## 404 2.463298882 -0.48724019 1.01499462 -0.2723291 1.19354259
## 405 4.408007629 -0.48724019 1.01499462 -0.2723291 1.19354259
## 406 7.476247076 -0.48724019 1.01499462 -0.2723291 1.19354259
## 407 1.988326078 -0.48724019 1.01499462 -0.2723291 0.90012973
## 408 0.969311483 -0.48724019 1.01499462 -0.2723291 0.90012973
## 409 0.440661113 -0.48724019 1.01499462 -0.2723291 0.36508275
## 410 1.258468834 -0.48724019 1.01499462 -0.2723291 0.36508275
## 411 5.524853484 -0.48724019 1.01499462 -0.2723291 0.36508275
## 412 1.213407163 -0.48724019 1.01499462 -0.2723291 0.36508275
## 413 1.766830989 -0.48724019 1.01499462 -0.2723291 0.36508275
## 414 2.911369543 -0.48724019 1.01499462 -0.2723291 0.36508275
## 415 4.898256758 -0.48724019 1.01499462 -0.2723291 1.19354259
## 416 1.682381045 -0.48724019 1.01499462 -0.2723291 1.07272553
## 417 0.839462719 -0.48724019 1.01499462 -0.2723291 1.07272553
## 418 2.595705326 -0.48724019 1.01499462 -0.2723291 1.07272553
## 419 8.128839131 -0.48724019 1.01499462 -0.2723291 1.07272553
## 420 0.953174847 -0.48724019 1.01499462 -0.2723291 1.40928733
## 421 0.868899291 -0.48724019 1.01499462 -0.2723291 1.40928733
## 422 0.396331868 -0.48724019 1.01499462 -0.2723291 1.40928733
## 423 0.980600153 -0.48724019 1.01499462 -0.2723291 0.51178918
## 424 0.399567334 -0.48724019 1.01499462 -0.2723291 0.51178918
## 425 0.602054210 -0.48724019 1.01499462 -0.2723291 0.25289548
## 426 1.423787970 -0.48724019 1.01499462 -0.2723291 1.07272553
## 427 1.003735531 -0.48724019 1.01499462 -0.2723291 0.25289548
## 428 3.958402360 -0.48724019 1.01499462 -0.2723291 1.07272553
## 429 0.436385137 -0.48724019 1.01499462 -0.2723291 1.07272553
## 430 0.665620696 -0.48724019 1.01499462 -0.2723291 1.07272553
## 431 0.567177918 -0.48724019 1.01499462 -0.2723291 0.25289548
## 432 0.749723028 -0.48724019 1.01499462 -0.2723291 0.25289548
## 433 0.329071860 -0.48724019 1.01499462 -0.2723291 0.25289548
## 434 0.228743373 -0.48724019 1.01499462 -0.2723291 1.36613839
## 435 1.197444914 -0.48724019 1.01499462 -0.2723291 1.36613839
## 436 0.877386138 -0.48724019 1.01499462 -0.2723291 1.59914271
## 437 1.256434316 -0.48724019 1.01499462 -0.2723291 1.59914271
## 438 1.344372005 -0.48724019 1.01499462 -0.2723291 1.59914271
## 439 1.170089364 -0.48724019 1.01499462 -0.2723291 1.59914271
## 440 0.671635895 -0.48724019 1.01499462 -0.2723291 1.59914271
## 441 2.143519126 -0.48724019 1.01499462 -0.2723291 1.59914271
## 442 0.710413811 -0.48724019 1.01499462 -0.2723291 1.59914271
## 443 0.238660196 -0.48724019 1.01499462 -0.2723291 1.59914271
## 444 0.738590145 -0.48724019 1.01499462 -0.2723291 1.59914271
## 445 1.068270448 -0.48724019 1.01499462 -0.2723291 1.59914271
## 446 0.820582390 -0.48724019 1.01499462 -0.2723291 1.59914271
## 447 0.310937908 -0.48724019 1.01499462 -0.2723291 1.59914271
## 448 0.733743341 -0.48724019 1.01499462 -0.2723291 1.59914271
## 449 0.664481366 -0.48724019 1.01499462 -0.2723291 1.36613839
## 450 0.454858563 -0.48724019 1.01499462 -0.2723291 1.36613839
## 451 0.360888236 -0.48724019 1.01499462 -0.2723291 1.36613839
## 452 0.212475366 -0.48724019 1.01499462 -0.2723291 1.36613839
## 453 0.171672232 -0.48724019 1.01499462 -0.2723291 1.36613839
## 454 0.538806271 -0.48724019 1.01499462 -0.2723291 1.36613839
## 455 0.685935651 -0.48724019 1.01499462 -0.2723291 1.36613839
## 456 0.132400217 -0.48724019 1.01499462 -0.2723291 1.36613839
## 457 0.122688009 -0.48724019 1.01499462 -0.2723291 1.36613839
## 458 0.533282845 -0.48724019 1.01499462 -0.2723291 1.36613839
## 459 0.481158489 -0.48724019 1.01499462 -0.2723291 1.36613839
## 460 0.370589982 -0.48724019 1.01499462 -0.2723291 1.36613839
## 461 0.139347806 -0.48724019 1.01499462 -0.2723291 1.36613839
## 462 0.009252575 -0.48724019 1.01499462 -0.2723291 1.36613839
## 463 0.353587222 -0.48724019 1.01499462 -0.2723291 1.36613839
## 464 0.256654638 -0.48724019 1.01499462 -0.2723291 1.36613839
## 465 0.491283414 -0.48724019 1.01499462 -0.2723291 0.86561057
## 466 -0.052307295 -0.48724019 1.01499462 -0.2723291 0.86561057
## 467 0.018770633 -0.48724019 1.01499462 -0.2723291 0.86561057
## 468 0.094024554 -0.48724019 1.01499462 -0.2723291 0.25289548
## 469 1.390700891 -0.48724019 1.01499462 -0.2723291 0.21837632
## 470 1.099985680 -0.48724019 1.01499462 -0.2723291 0.21837632
## 471 0.085480740 -0.48724019 1.01499462 -0.2723291 0.21837632
## 472 0.049396526 -0.48724019 1.01499462 -0.2723291 -0.19585359
## 473 -0.005213430 -0.48724019 1.01499462 -0.2723291 0.21837632
## 474 0.120137304 -0.48724019 1.01499462 -0.2723291 0.51178918
## 475 0.516449822 -0.48724019 1.01499462 -0.2723291 0.25289548
## 476 0.323150830 -0.48724019 1.01499462 -0.2723291 0.25289548
## 477 0.146239592 -0.48724019 1.01499462 -0.2723291 0.51178918
## 478 1.326491497 -0.48724019 1.01499462 -0.2723291 0.51178918
## 479 0.769568300 -0.48724019 1.01499462 -0.2723291 0.51178918
## 480 1.246308229 -0.48724019 1.01499462 -0.2723291 0.51178918
## 481 0.256987136 -0.48724019 1.01499462 -0.2723291 -0.19585359
## 482 0.243520951 -0.48724019 1.01499462 -0.2723291 -0.19585359
## 483 0.246192564 -0.48724019 1.01499462 -0.2723291 -0.19585359
## 484 -0.092441945 -0.48724019 1.01499462 -0.2723291 -0.19585359
## 485 -0.143573456 -0.48724019 1.01499462 -0.2723291 0.24426569
## 486 0.006992516 -0.48724019 1.01499462 -0.2723291 0.24426569
## 487 0.241610829 -0.48724019 1.01499462 -0.2723291 0.24426569
## 488 0.142084524 -0.48724019 1.01499462 -0.2723291 0.24426569
## 489 -0.402562972 -0.48724019 2.42017014 -0.2723291 0.46864023
## 490 -0.398783418 -0.48724019 2.42017014 -0.2723291 0.46864023
## 491 -0.395982758 -0.48724019 2.42017014 -0.2723291 0.46864023
## 492 -0.407808541 -0.48724019 2.42017014 -0.2723291 0.46864023
## 493 -0.407159820 -0.48724019 2.42017014 -0.2723291 0.46864023
## 494 -0.399952975 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 495 -0.387599381 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 496 -0.399292629 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 497 -0.386433311 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 498 -0.388900310 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 499 -0.392302024 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 500 -0.399427488 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 501 -0.394015670 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 502 -0.412820431 -0.48724019 0.11562398 -0.2723291 0.15796779
## 503 -0.414838673 -0.48724019 0.11562398 -0.2723291 0.15796779
## 504 -0.413037834 -0.48724019 0.11562398 -0.2723291 0.15796779
## 505 -0.407360947 -0.48724019 0.11562398 -0.2723291 0.15796779
## 506 -0.414589880 -0.48724019 0.11562398 -0.2723291 0.15796779
## rm age dis rad tax
## 1 0.413262920 -0.119894767 0.1400749840 -0.9818712 -0.66594918
## 2 0.194082387 0.366803426 0.5566090496 -0.8670245 -0.98635338
## 3 1.281445551 -0.265548971 0.5566090496 -0.8670245 -0.98635338
## 4 1.015297761 -0.809087830 1.0766711351 -0.7521778 -1.10502160
## 5 1.227362043 -0.510674339 1.0766711351 -0.7521778 -1.10502160
## 6 0.206891639 -0.350809969 1.0766711351 -0.7521778 -1.10502160
## 7 -0.388026950 -0.070159185 0.8384142195 -0.5224844 -0.57694801
## 8 -0.160306916 0.977840575 1.0236248974 -0.5224844 -0.57694801
## 9 -0.930285282 1.116389695 1.0861216287 -0.5224844 -0.57694801
## 10 -0.399412952 0.615481336 1.3283202075 -0.5224844 -0.57694801
## 11 0.131459378 0.913894827 1.2117799501 -0.5224844 -0.57694801
## 12 -0.392296701 0.508905089 1.1547920492 -0.5224844 -0.57694801
## 13 -0.563086727 -1.050660656 0.7863652700 -0.5224844 -0.57694801
## 14 -0.477691714 -0.240681180 0.4333252240 -0.6373311 -0.60068166
## 15 -0.268473932 0.565745754 0.3166899868 -0.6373311 -0.60068166
## 16 -0.641365488 -0.428965883 0.3341187865 -0.6373311 -0.60068166
## 17 -0.497617217 -1.395257187 0.3341187865 -0.6373311 -0.60068166
## 18 -0.419338455 0.466274590 0.2198105553 -0.6373311 -0.60068166
## 19 -1.179354069 -1.135921653 0.0006920764 -0.6373311 -0.60068166
## 20 -0.793653261 0.032864520 0.0006920764 -0.6373311 -0.60068166
## 21 -1.017103545 1.048891406 0.0013569352 -0.6373311 -0.60068166
## 22 -0.454919710 0.732715207 0.1031753182 -0.6373311 -0.60068166
## 23 -0.203004422 0.821528746 0.0863638874 -0.6373311 -0.60068166
## 24 -0.671253743 1.116389695 0.1425444597 -0.6373311 -0.60068166
## 25 -0.513272969 0.906789743 0.2871037683 -0.6373311 -0.60068166
## 26 -0.975829289 0.608376252 0.3132232229 -0.6373311 -0.60068166
## 27 -0.671253743 0.771793164 0.4212152950 -0.6373311 -0.60068166
## 28 -0.338213193 0.718505041 0.3126533438 -0.6373311 -0.60068166
## 29 0.299402903 0.917447368 0.3132707128 -0.6373311 -0.60068166
## 30 0.554164692 0.665216917 0.2108349609 -0.6373311 -0.60068166
## 31 -0.813578764 0.906789743 0.2079855659 -0.6373311 -0.60068166
## 32 -0.302631937 1.116389695 0.1804414138 -0.6373311 -0.60068166
## 33 -0.476268464 0.476932215 0.0925850666 -0.6373311 -0.60068166
## 34 -0.830657767 0.938762618 -0.0037244859 -0.6373311 -0.60068166
## 35 -0.268473932 1.006260907 -0.0167367233 -0.6373311 -0.60068166
## 36 -0.500463717 -0.013318520 -0.2064589434 -0.5224844 -0.76681717
## 37 -0.631402737 -0.254891346 -0.1981007179 -0.5224844 -0.76681717
## 38 -0.618593485 -0.961847117 0.0660856927 -0.5224844 -0.76681717
## 39 -0.453496460 -1.363284313 0.0248169544 -0.5224844 -0.76681717
## 40 0.441727925 -1.661697804 0.7627152911 -0.7521778 -0.92701927
## 41 1.052302267 -1.874850298 0.7627152911 -0.7521778 -0.92701927
## 42 0.690796712 -2.333128159 0.9145880470 -0.7521778 -1.03975408
## 43 -0.164576667 -2.201684121 0.9145880470 -0.7521778 -1.03975408
## 44 -0.104800158 -2.205236663 0.9145880470 -0.7521778 -1.03975408
## 45 -0.306901688 -1.015135240 0.9145880470 -0.7521778 -1.03975408
## 46 -0.857699521 -1.235392817 0.6199131095 -0.7521778 -1.03975408
## 47 -0.709681499 -1.253155525 0.6199131095 -0.7521778 -1.03975408
## 48 -0.362408446 0.601271169 0.8996287230 -0.7521778 -1.03975408
## 49 -1.260479332 0.949420242 0.9853955139 -0.7521778 -1.03975408
## 50 -0.971559538 -0.233576097 1.0887810641 -0.7521778 -1.03975408
## 51 -0.457766211 -0.812640371 1.4340327636 -0.6373311 -0.98041997
## 52 -0.241432178 -0.198050682 1.4340327636 -0.6373311 -0.98041997
## 53 0.322174907 -1.686565595 1.4340327636 -0.6373311 -0.98041997
## 54 -0.407952453 -1.675907970 1.4340327636 -0.6373311 -0.98041997
## 55 -0.564509977 -0.745142082 1.6738568465 -0.7521778 0.36053095
## 56 1.372533565 -1.658145262 2.3277455193 -0.5224844 -1.08128796
## 57 0.139998879 -1.167894527 2.5609210138 -0.8670245 -0.56508119
## 58 0.756266222 -0.997372532 2.1511780064 -0.5224844 -0.90328562
## 59 -0.198734672 -1.398809729 1.9089794276 -0.1779443 -0.73715011
## 60 -0.509003218 -0.759352248 1.4897384367 -0.1779443 -0.73715011
## 61 -0.773727758 -0.084369352 1.6290738544 -0.1779443 -0.73715011
## 62 -0.453496460 0.881921953 1.4358373805 -0.1779443 -0.73715011
## 63 0.243896145 -0.027528687 1.6291213443 -0.1779443 -0.73715011
## 64 0.679410711 -0.894348827 1.9878601804 -0.1779443 -0.73715011
## 65 1.166162284 -0.322389636 2.5776849546 -0.7521778 -1.14062207
## 66 0.007636609 -1.803799466 1.3375332514 -0.6373311 -0.42267932
## 67 -0.708258248 -1.331311439 1.3375332514 -0.6373311 -0.42267932
## 68 -0.578742479 -1.675907970 1.2836321952 -0.6373311 -0.37521203
## 69 -0.982945540 -1.128816570 1.2836321952 -0.6373311 -0.37521203
## 70 -0.568779727 -1.263813149 1.2836321952 -0.6373311 -0.37521203
## 71 0.188389387 -2.201684121 0.7086717651 -0.6373311 -0.61254848
## 72 -0.460612711 -1.814457091 0.7086717651 -0.6373311 -0.61254848
## 73 -0.312594689 -2.159053622 0.7086717651 -0.6373311 -0.61254848
## 74 -0.056409650 -2.215894287 0.7086717651 -0.6373311 -0.61254848
## 75 -0.016558644 -2.222999370 0.2167712006 -0.5224844 -0.06074124
## 76 0.001943608 -0.837508162 0.3360183832 -0.5224844 -0.06074124
## 77 -0.008019143 0.210491598 0.1221237952 -0.5224844 -0.06074124
## 78 -0.205850923 -0.809087830 0.1403124336 -0.5224844 -0.06074124
## 79 -0.074911903 -0.528437047 0.5789293108 -0.5224844 -0.06074124
## 80 -0.584435480 -1.135921653 0.3360183832 -0.5224844 -0.06074124
## 81 0.629596953 -1.246050442 0.7625253315 -0.6373311 -0.75495035
## 82 0.475885930 0.064837394 0.7625253315 -0.6373311 -0.75495035
## 83 0.024715612 -1.292233482 0.7625253315 -0.6373311 -0.75495035
## 84 -0.167423167 -0.777114956 0.7625253315 -0.6373311 -0.75495035
## 85 0.148538381 -0.730931915 0.4674704746 -0.7521778 -0.95668632
## 86 0.491541682 -0.443176049 0.3051974268 -0.7521778 -0.95668632
## 87 -0.383757200 -0.833955621 0.3002109855 -0.7521778 -0.95668632
## 88 -0.232892677 -0.418308258 -0.0225304932 -0.7521778 -0.95668632
## 89 1.028107013 0.629691502 -0.1773001341 -0.8670245 -0.82021787
## 90 1.130581029 -0.194498140 -0.1807194081 -0.8670245 -0.82021787
## 91 0.188389387 -0.087921893 -0.3337319220 -0.8670245 -0.82021787
## 92 0.171310384 0.189176348 -0.3338269018 -0.8670245 -0.82021787
## 93 0.223970642 -0.531989588 -0.0613297558 -0.6373311 -0.82021787
## 94 -0.104800158 -1.409467353 -0.0613297558 -0.6373311 -0.82021787
## 95 -0.050716649 0.309962761 -0.0855021237 -0.6373311 -0.82021787
## 96 0.484425431 -0.382782843 -0.1423950448 -0.8670245 -0.78461740
## 97 -0.173116168 0.036417061 -0.1423950448 -0.8670245 -0.78461740
## 98 2.539598741 0.263779721 -0.1423950448 -0.8670245 -0.78461740
## 99 2.185209437 -1.125264029 -0.1423950448 -0.8670245 -0.78461740
## 100 1.610216351 -0.215813389 -0.1423950448 -0.8670245 -0.78461740
## 101 0.629596953 0.402328842 -0.4830877123 -0.5224844 -0.14380900
## 102 0.706452465 0.096810268 -0.4459031069 -0.5224844 -0.14380900
## 103 0.171310384 0.597718628 -0.5130538501 -0.5224844 -0.14380900
## 104 -0.210120673 0.668769459 -0.5130538501 -0.5224844 -0.14380900
## 105 -0.167423167 0.761135540 -0.6525317376 -0.5224844 -0.14380900
## 106 -0.617170235 0.999155824 -0.8016975682 -0.5224844 -0.14380900
## 107 -0.638518988 0.828633829 -0.7522605641 -0.5224844 -0.14380900
## 108 -0.224353176 0.590613545 -0.7943366310 -0.5224844 -0.14380900
## 109 0.269514649 1.013365990 -0.6468804374 -0.5224844 -0.14380900
## 110 -0.079181654 0.803766038 -0.5935967501 -0.5224844 -0.14380900
## 111 -0.127572161 -0.503569256 -0.4830877123 -0.5224844 -0.14380900
## 112 0.612517950 0.462722049 -0.5307200994 -0.4076377 0.14099473
## 113 -0.528928721 0.864159245 -0.6846349217 -0.4076377 0.14099473
## 114 -0.274166933 0.952972784 -0.5922195425 -0.4076377 0.14099473
## 115 -0.043600398 0.555088129 -0.7306526517 -0.4076377 0.14099473
## 116 -0.507579968 0.697189791 -0.6325384823 -0.4076377 0.14099473
## 117 -0.154613915 0.139440767 -0.5057404029 -0.4076377 0.14099473
## 118 -0.375217698 0.498247464 -0.4975246471 -0.4076377 0.14099473
## 119 -0.587281980 0.160756016 -0.6256999342 -0.4076377 0.14099473
## 120 -0.787960260 -0.119894767 -0.4919208369 -0.4076377 0.14099473
## 121 -0.590128481 0.039969603 -0.7300827727 -0.8670245 -1.30675758
## 122 -0.399412952 0.551535588 -0.7587191929 -0.8670245 -1.30675758
## 123 -0.460612711 0.864159245 -0.8111955516 -0.8670245 -1.30675758
## 124 -0.610053984 1.009813449 -0.8788686839 -0.8670245 -1.30675758
## 125 -0.577319229 0.967182950 -0.8494724251 -0.8670245 -1.30675758
## 126 -0.425031456 0.704294875 -0.8558360740 -0.8670245 -1.30675758
## 127 -0.955903786 0.960077867 -0.9677698093 -0.8670245 -1.30675758
## 128 -0.842043769 0.974288033 -0.9530004450 -0.6373311 0.17066179
## 129 0.208314890 1.073759197 -0.9415078850 -0.6373311 0.17066179
## 130 -0.921745781 0.928104993 -0.8620097633 -0.6373311 0.17066179
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## 355 -0.884741275 -1.658145262 3.2248775491 -0.6373311 -0.44047956
## 356 -0.496193967 -1.743406260 3.2248775491 -0.6373311 -0.44047956
## 357 -0.103376907 1.024023615 -0.7944316108 1.6596029 1.52941294
## 358 0.157077882 0.796660955 -0.6125452271 1.6596029 1.52941294
## 359 -0.224353176 0.526667797 -0.5092546567 1.6596029 1.52941294
## 360 -0.245701929 0.452064424 -0.6106931203 1.6596029 1.52941294
## 361 0.161347633 0.690084708 -0.6063715378 1.6596029 1.52941294
## 362 -0.047870149 0.800213497 -0.7121315839 1.6596029 1.52941294
## 363 -1.313139590 0.981393116 -0.8032647354 1.6596029 1.52941294
## 364 -0.685486245 0.725610124 -0.8977221812 1.6596029 1.52941294
## 365 3.551529643 0.508905089 -0.8977221812 1.6596029 1.52941294
## 366 -3.876413226 0.686532167 -1.0361552904 1.6596029 1.52941294
## 367 -1.881016425 0.810871121 -0.9700968153 1.6596029 1.52941294
## 368 -3.446591661 1.116389695 -1.0848799457 1.6596029 1.52941294
## 369 -1.871053674 1.116389695 -1.1694594886 1.6596029 1.52941294
## 370 0.566973944 1.002708366 -1.1579669285 1.6596029 1.52941294
## 371 1.040916265 1.027576156 -1.2312438711 1.6596029 1.52941294
## 372 -0.097683907 1.116389695 -1.2470580136 1.6596029 1.52941294
## 373 -0.583012230 0.746925373 -1.2658165310 1.6596029 1.52941294
## 374 -1.962141687 1.116389695 -1.2446360278 1.6596029 1.52941294
## 375 -3.055197852 1.116389695 -1.2623022771 1.6596029 1.52941294
## 376 1.463621579 1.041786323 -1.1771528552 1.6596029 1.52941294
## 377 0.518583436 0.878369411 -1.1635707388 1.6596029 1.52941294
## 378 0.724954717 1.073759197 -1.1573495596 1.6596029 1.52941294
## 379 0.135729129 0.981393116 -1.1440048928 1.6596029 1.52941294
## 380 -0.087721155 1.116389695 -1.1440048928 1.6596029 1.52941294
## 381 0.972600255 0.828633829 -1.1295679579 1.6596029 1.52941294
## 382 0.370565414 1.084416821 -1.0807958128 1.6596029 1.52941294
## 383 -1.065494052 1.116389695 -1.0517319833 1.6596029 1.52941294
## 384 -1.088266056 1.116389695 -1.0741947142 1.6596029 1.52941294
## 385 -2.727850303 0.803766038 -1.1186452769 1.6596029 1.52941294
## 386 -1.434115858 1.048891406 -1.1250089259 1.6596029 1.52941294
## 387 -2.323647242 1.116389695 -1.1054905698 1.6596029 1.52941294
## 388 -1.828356167 0.743372832 -1.0811757321 1.6596029 1.52941294
## 389 -1.999146193 1.116389695 -1.0474104008 1.6596029 1.52941294
## 390 -1.273288584 1.077311738 -0.9815893753 1.6596029 1.52941294
## 391 -0.813578764 1.009813449 -0.8873693792 1.6596029 1.52941294
## 392 -0.332520192 0.494694923 -0.7727762084 1.6596029 1.52941294
## 393 -1.777119159 1.009813449 -0.9616910999 1.6596029 1.52941294
## 394 -0.130418661 0.853501620 -0.9516232374 1.6596029 1.52941294
## 395 -0.565933227 0.928104993 -0.9559448199 1.6596029 1.52941294
## 396 0.265244898 1.073759197 -0.9827291333 1.6596029 1.52941294
## 397 0.171310384 0.974288033 -1.0059517029 1.6596029 1.52941294
## 398 -0.765188257 1.077311738 -1.0265623271 1.6596029 1.52941294
## 399 -1.183623820 1.116389695 -1.0948528283 1.6596029 1.52941294
## 400 -0.615746985 0.327725469 -1.0897239172 1.6596029 1.52941294
## 401 -0.423608206 1.116389695 -1.0477428302 1.6596029 1.52941294
## 402 0.083068871 1.116389695 -1.0547238481 1.6596029 1.52941294
## 403 0.169887134 1.116389695 -1.0239028917 1.6596029 1.52941294
## 404 -1.331641842 0.974288033 -0.9936043244 1.6596029 1.52941294
## 405 -1.072610303 0.597718628 -1.0389097056 1.6596029 1.52941294
## 406 -0.856276271 1.116389695 -1.1253413553 1.6596029 1.52941294
## 407 -3.055197852 1.116389695 -1.2427839210 1.6596029 1.52941294
## 408 -0.963020037 1.116389695 -1.1919222195 1.6596029 1.52941294
## 409 -0.950210785 1.041786323 -1.1114268095 1.6596029 1.52941294
## 410 0.807503230 1.116389695 -1.1062978984 1.6596029 1.52941294
## 411 -0.750955755 1.116389695 -1.1312301050 1.6596029 1.52941294
## 412 0.529969438 1.116389695 -1.0768541496 1.6596029 1.52941294
## 413 -2.357805247 1.116389695 -1.0643168114 1.6596029 1.52941294
## 414 -1.607752384 1.116389695 -1.0474578907 1.6596029 1.52941294
## 415 -2.512939520 1.116389695 -1.0147848276 1.6596029 1.52941294
## 416 0.212584640 1.116389695 -0.9309651233 1.6596029 1.52941294
## 417 0.707875715 0.789555872 -0.9381835908 1.6596029 1.52941294
## 418 -1.395688102 0.729162665 -1.0198662487 1.6596029 1.52941294
## 419 -0.466305712 1.116389695 -0.9462093868 1.6596029 1.52941294
## 420 0.767652224 0.281542429 -0.9502935197 1.6596029 1.52941294
## 421 0.179849885 1.116389695 -0.9194725633 1.6596029 1.52941294
## 422 -0.396566452 0.949420242 -0.9120166463 1.6596029 1.52941294
## 423 -0.906090028 0.675874542 -0.8756393696 1.6596029 1.52941294
## 424 -0.258511181 0.587061003 -0.8421114879 1.6596029 1.52941294
## 425 -1.024219796 0.071942477 -0.8223081923 1.6596029 1.52941294
## 426 -0.553123975 0.952972784 -0.8953951752 1.6596029 1.52941294
## 427 -0.637095738 -0.315284553 -0.8536040479 1.6596029 1.52941294
## 428 -0.117609410 0.359698343 -0.9175729666 1.6596029 1.52941294
## 429 -0.130418661 0.338383094 -0.8830477967 1.6596029 1.52941294
## 430 0.135729129 0.960077867 -0.8675660836 1.6596029 1.52941294
## 431 0.090185122 0.622586419 -0.8274371034 1.6596029 1.52941294
## 432 0.780461476 0.913894827 -0.8105781827 1.6596029 1.52941294
## 433 0.199775388 0.221149222 -0.7572944954 1.6596029 1.52941294
## 434 0.215431141 0.686532167 -0.7024911307 1.6596029 1.52941294
## 435 -0.109069908 0.938762618 -0.7469416934 1.6596029 1.52941294
## 436 0.490118432 0.924552451 -0.7932443629 1.6596029 1.52941294
## 437 0.251012396 0.878369411 -0.8512295520 1.6596029 1.52941294
## 438 -0.188771920 1.116389695 -0.8932106390 1.6596029 1.52941294
## 439 -0.497617217 0.686532167 -0.9376612017 1.6596029 1.52941294
## 440 -0.935978283 0.899684660 -0.9392758589 1.6596029 1.52941294
## 441 -0.664137492 0.846396537 -0.9160057994 1.6596029 1.52941294
## 442 0.172733634 1.016918532 -0.8215483536 1.6596029 1.52941294
## 443 -0.093414156 1.116389695 -0.8501847738 1.6596029 1.52941294
## 444 0.285170401 1.116389695 -0.8627221120 1.6596029 1.52941294
## 445 -0.612900484 0.995603282 -0.9020437636 1.6596029 1.52941294
## 446 0.248165896 0.931657534 -0.8582105699 1.6596029 1.52941294
## 447 0.080222370 0.988498199 -0.8182715493 1.6596029 1.52941294
## 448 -0.047870149 0.995603282 -0.7584342534 1.6596029 1.52941294
## 449 -0.141804663 1.070206655 -0.7282306659 1.6596029 1.52941294
## 450 0.188389387 1.055996489 -0.7646079427 1.6596029 1.52941294
## 451 0.660908458 0.853501620 -0.6987869171 1.6596029 1.52941294
## 452 0.527122938 1.052443947 -0.6837801032 1.6596029 1.52941294
## 453 0.017599361 0.825081288 -0.6776064140 1.6596029 1.52941294
## 454 1.577481596 1.091521905 -0.6374774338 1.6596029 1.52941294
## 455 0.631020203 0.906789743 -0.6168668096 1.6596029 1.52941294
## 456 0.342100410 0.636796585 -0.6455032298 1.6596029 1.52941294
## 457 -0.439263958 0.686532167 -0.5767378294 1.6596029 1.52941294
## 458 -0.496193967 0.416539008 -0.4824228534 1.6596029 1.52941294
## 459 0.023292362 0.537325421 -0.4805707466 1.6596029 1.52941294
## 460 -0.289822685 0.562193212 -0.5117241325 1.6596029 1.52941294
## 461 0.592592447 0.761135540 -0.5687120333 1.6596029 1.52941294
## 462 0.130036128 0.704294875 -0.5831489682 1.6596029 1.52941294
## 463 0.046064365 0.512457630 -0.5036983364 1.6596029 1.52941294
## 464 0.325021407 0.757582998 -0.4717851119 1.6596029 1.52941294
## 465 -0.107646658 -0.112789684 -0.3949464255 1.6596029 1.52941294
## 466 -0.748109254 -0.723826832 -0.3459843207 1.6596029 1.52941294
## 467 -0.473421963 0.572850837 -0.4385896596 1.6596029 1.52941294
## 468 -0.400836202 0.920999910 -0.5958762661 1.6596029 1.52941294
## 469 -0.510426469 0.086152643 -0.4210658801 1.6596029 1.52941294
## 470 -0.813578764 -0.421860800 -0.4612898402 1.6596029 1.52941294
## 471 -0.167423167 0.547983046 -0.3617034834 1.6596029 1.52941294
## 472 -0.079181654 0.786003330 -0.3304076278 1.6596029 1.52941294
## 473 0.216854391 0.228254306 -0.4267171803 1.6596029 1.52941294
## 474 0.989679257 -0.034633770 -0.5993905200 1.6596029 1.52941294
## 475 -1.220628326 0.952972784 -0.6483526248 1.6596029 1.52941294
## 476 -0.174539418 1.024023615 -0.7546350600 1.6596029 1.52941294
## 477 0.283747151 0.889027036 -0.7074775720 1.6596029 1.52941294
## 478 -1.395688102 1.020471073 -0.8046419430 1.6596029 1.52941294
## 479 -0.141804663 0.999155824 -0.7714939807 1.6596029 1.52941294
## 480 -0.079181654 0.690084708 -0.8756393696 1.6596029 1.52941294
## 481 -0.060679401 -0.137657475 -0.1761128861 1.6596029 1.52941294
## 482 0.662331708 0.224701764 -0.2200410597 1.6596029 1.52941294
## 483 1.104962525 0.299305137 -0.1825715149 1.6596029 1.52941294
## 484 -0.743839504 -1.004477616 0.1440166471 1.6596029 1.52941294
## 485 -0.588705230 -0.947636951 -0.0337381137 1.6596029 1.52941294
## 486 0.038948114 -0.592382795 0.0933923952 1.6596029 1.52941294
## 487 -0.242855428 0.398776300 -0.1183176566 1.6596029 1.52941294
## 488 -0.540314723 -0.546199754 -0.3052379716 1.6596029 1.52941294
## 489 -1.182200570 0.857054162 -0.9375187319 -0.6373311 1.79641644
## 490 -1.239130578 1.055996489 -0.9686246278 -0.6373311 1.79641644
## 491 -1.695993897 1.045338864 -0.9367114033 -0.6373311 1.79641644
## 492 -0.429301206 1.073759197 -0.9151034909 -0.6373311 1.79641644
## 493 -0.429301206 0.530220338 -0.8002728706 -0.6373311 1.79641644
## 494 -0.822118266 -0.517779422 -0.6711952751 -0.4076377 -0.10227512
## 495 -0.510426469 -0.922769160 -0.6711952751 -0.4076377 -0.10227512
## 496 -0.874778524 -1.413019895 -0.4732098094 -0.4076377 -0.10227512
## 497 -1.273288584 0.153650933 -0.4732098094 -0.4076377 -0.10227512
## 498 -0.698295497 0.071942477 -0.4285217972 -0.4076377 -0.10227512
## 499 -0.378064199 -0.116342226 -0.6581830377 -0.4076377 -0.10227512
## 500 -1.018526795 0.174966182 -0.6625521101 -0.4076377 -0.10227512
## 501 -0.366678197 0.395223759 -0.6158695213 -0.4076377 -0.10227512
## 502 0.438881424 0.018654354 -0.6251775451 -0.9818712 -0.80241764
## 503 -0.234315927 0.288647512 -0.7159307773 -0.9818712 -0.80241764
## 504 0.983986256 0.796660955 -0.7729186782 -0.9818712 -0.80241764
## 505 0.724954717 0.736267749 -0.6677760011 -0.9818712 -0.80241764
## 506 -0.362408446 0.434301716 -0.6126402069 -0.9818712 -0.80241764
## ptratio black lstat medv
## 1 -1.45755797 0.440615895 -1.074498970 0.159527789
## 2 -0.30279450 0.440615895 -0.491952525 -0.101423917
## 3 -0.30279450 0.396035074 -1.207532413 1.322937477
## 4 0.11292035 0.415751408 -1.360170785 1.181588636
## 5 0.11292035 0.440615895 -1.025486649 1.486032293
## 6 0.11292035 0.410165113 -1.042290874 0.670558212
## 7 -1.50374851 0.426376321 -0.031236706 0.039924923
## 8 -1.50374851 0.440615895 0.909799859 0.496590409
## 9 -1.50374851 0.328123258 2.419379350 -0.655946292
## 10 -1.50374851 0.328999540 0.622727693 -0.394994586
## 11 -1.50374851 0.392639483 1.091845624 -0.819041108
## 12 -1.50374851 0.440615895 0.086392864 -0.394994586
## 13 -1.50374851 0.370513376 0.428078760 -0.090550930
## 14 1.17530274 0.440615895 -0.615183504 -0.231899770
## 15 1.17530274 0.255720500 -0.335113097 -0.471105500
## 16 1.17530274 0.426595391 -0.585776111 -0.286264709
## 17 1.17530274 0.330533033 -0.850442645 0.061670899
## 18 1.17530274 0.329437681 0.282442149 -0.547216415
## 19 1.17530274 -0.741378303 -0.134862757 -0.253645746
## 20 1.17530274 0.375442459 -0.192277190 -0.471105500
## 21 1.17530274 0.217930861 1.171665689 -0.971262937
## 22 1.17530274 0.392749018 0.164812578 -0.318883672
## 23 1.17530274 0.440615895 0.849584722 -0.797295133
## 24 1.17530274 0.414765591 1.012025558 -0.873406047
## 25 1.17530274 0.412465352 0.510699530 -0.753803182
## 26 1.17530274 -0.583319029 0.540106923 -0.938643973
## 27 1.17530274 0.221326452 0.302047077 -0.645073304
## 28 1.17530274 -0.550896614 0.647934029 -0.840787084
## 29 1.17530274 0.342472368 0.020576319 -0.449359525
## 30 1.17530274 0.258020739 -0.094252548 -0.166661844
## 31 1.17530274 0.038293155 1.392921311 -1.069119826
## 32 1.17530274 0.219683424 0.054184768 -0.873406047
## 33 1.17530274 -1.359047220 2.108501199 -1.014754888
## 34 1.17530274 0.022958229 0.797771697 -1.025627875
## 35 1.17530274 -1.186967442 1.076441751 -0.982135924
## 36 0.34387304 0.440615895 -0.416333515 -0.394994586
## 37 0.34387304 0.228774844 -0.174072614 -0.275391721
## 38 0.34387304 0.440615895 -0.543765550 -0.166661844
## 39 0.34387304 0.402607185 -0.353317674 0.235638703
## 40 -0.07184181 0.426704926 -1.166922204 0.898890955
## 41 -0.07184181 0.426595391 -1.494604580 1.344683452
## 42 -0.25660396 0.314759966 -1.094103899 0.442225470
## 43 -0.25660396 0.292414788 -0.958269752 0.300876629
## 44 -0.25660396 0.413889309 -0.730012370 0.235638703
## 45 -0.25660396 0.358354970 -0.434538092 -0.144915868
## 46 -0.25660396 0.440615895 -0.342114857 -0.351502635
## 47 -0.25660396 0.440615895 0.209623843 -0.275391721
## 48 -0.25660396 0.395049257 0.860787538 -0.645073304
## 49 -0.25660396 0.440615895 2.542610329 -0.884279035
## 50 -0.25660396 0.440615895 0.496696010 -0.340629648
## 51 -0.76469989 0.425938180 0.111599201 -0.308010684
## 52 -0.76469989 0.408522085 -0.451342316 -0.221026782
## 53 -0.76469989 0.440615895 -1.032488409 0.268257666
## 54 -0.76469989 0.440615895 -0.591377519 0.094289862
## 55 1.22149328 0.440615895 0.300646725 -0.394994586
## 56 -0.25660396 0.429990982 -1.098304955 1.399048391
## 57 -0.53374719 0.440615895 -0.963871160 0.235638703
## 58 -1.54993904 0.396801820 -1.218735230 0.985874857
## 59 0.57482574 0.372485009 -0.811232788 0.083416874
## 60 0.57482574 0.440615895 -0.480749709 -0.318883672
## 61 0.57482574 0.421009097 0.069588640 -0.416740562
## 62 0.57482574 0.234470674 0.250234052 -0.710311231
## 63 0.57482574 0.440615895 -0.829437365 -0.036185991
## 64 0.57482574 0.426157250 -0.441539852 0.268257666
## 65 0.06672981 0.400526017 -0.644590896 1.138096685
## 66 -1.08803366 0.440615895 -1.117909883 0.105162850
## 67 -1.08803366 0.440615895 -0.337913801 -0.340629648
## 68 0.20530143 0.433057967 -0.637589136 -0.057931966
## 69 0.20530143 0.440615895 0.061186528 -0.558089402
## 70 0.20530143 0.440615895 -0.540964846 -0.177534831
## 71 0.34387304 0.296358054 -0.830837717 0.181273764
## 72 0.34387304 0.221983663 -0.388326475 -0.090550930
## 73 0.34387304 0.375004318 -0.998879961 0.029051936
## 74 0.34387304 0.224502972 -0.716008850 0.094289862
## 75 0.11292035 0.418927928 -0.822435605 0.170400776
## 76 0.11292035 0.290881295 -0.519959566 -0.123169893
## 77 0.11292035 0.186056121 -0.095652900 -0.275391721
## 78 0.11292035 0.331737920 -0.333712745 -0.188407819
## 79 0.11292035 0.325603949 -0.043839875 -0.144915868
## 80 0.11292035 0.431414939 -0.497553933 -0.242772758
## 81 0.25149196 0.440615895 -1.031088057 0.594447298
## 82 0.25149196 0.426704926 -0.760820115 0.148654801
## 83 0.25149196 0.440615895 -0.830837717 0.246511690
## 84 0.25149196 0.372046868 -0.720209906 0.039924923
## 85 0.02053927 0.440615895 -0.424735627 0.148654801
## 86 0.02053927 0.390229709 -0.857444405 0.442225470
## 87 0.02053927 0.430648193 0.028978431 -0.003567028
## 88 0.02053927 0.421447237 -0.589977167 -0.036185991
## 89 -0.30279450 0.440615895 -1.001680665 0.116035838
## 90 -0.30279450 0.431414939 -0.973673624 0.670558212
## 91 -0.30279450 0.388915287 -0.538164142 0.007305960
## 92 -0.30279450 0.403921608 -0.623585616 -0.057931966
## 93 -0.11803234 0.419913745 -0.629187024 0.039924923
## 94 -0.11803234 0.434372389 -0.902255670 0.268257666
## 95 -0.11803234 0.440615895 -0.288901480 -0.210153795
## 96 -0.21041342 0.014304949 -0.840640181 0.637939249
## 97 -0.21041342 0.385081555 -0.183875078 -0.123169893
## 98 -0.21041342 0.440615895 -1.182326077 1.757856987
## 99 -0.21041342 0.403702537 -1.271948607 2.312379361
## 100 -0.21041342 0.440615895 -0.905056374 1.159842661
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## 381 0.80577843 0.440615895 0.638131565 -1.319198544
## 382 0.80577843 0.440615895 1.180067802 -1.264833606
## 383 0.80577843 0.440615895 1.532956514 -1.221341655
## 384 0.80577843 0.440615895 1.667390309 -1.112611777
## 385 0.80577843 -0.775991422 2.517403992 -1.493166348
## 386 0.80577843 0.440615895 2.542610329 -1.667134152
## 387 0.80577843 0.440615895 2.188321265 -1.308325557
## 388 0.80577843 0.440615895 2.707851869 -1.645388177
## 389 0.80577843 0.177950518 2.516003640 -1.340944520
## 390 0.80577843 0.440615895 1.147859705 -1.199595679
## 391 0.80577843 0.413560704 0.624128045 -0.808168120
## 392 0.80577843 0.237756730 0.855186130 0.072543887
## 393 0.80577843 0.440615895 1.824229736 -1.395309459
## 394 0.80577843 0.440615895 0.352459751 -0.949516961
## 395 0.80577843 0.440615895 0.517701290 -1.069119826
## 396 0.80577843 0.386724583 0.625528397 -1.025627875
## 397 0.80577843 0.440615895 0.940607604 -1.090865802
## 398 0.80577843 0.398992524 1.017626966 -1.525785311
## 399 0.80577843 0.440615895 2.511802584 -1.906339882
## 400 0.80577843 -0.202793791 2.424980758 -1.764991042
## 401 0.80577843 0.440615895 1.976868108 -1.841101956
## 402 0.80577843 0.440615895 1.073641047 -1.667134152
## 403 0.80577843 0.212892242 1.072240695 -1.134357753
## 404 0.80577843 0.440615895 0.996621685 -1.547531287
## 405 0.80577843 -0.298089403 2.062289582 -1.525785311
## 406 0.80577843 0.309940417 1.446134688 -1.906339882
## 407 0.80577843 0.148376017 1.496547361 -1.156103728
## 408 0.80577843 -0.269281649 -0.073247267 0.583574310
## 409 0.80577843 -0.460420549 1.925055083 -0.579835378
## 410 0.80577843 -1.942212553 0.998022037 0.540082359
## 411 0.80577843 -3.878356510 -0.356118378 -0.819041108
## 412 0.80577843 -3.522914830 1.199672730 -0.579835378
## 413 0.80577843 -3.591483857 3.041135653 -0.503724464
## 414 0.80577843 -1.595971828 1.040032598 -0.677692268
## 415 0.80577843 -2.939968567 3.406627533 -1.688880128
## 416 0.80577843 -3.608352275 2.296148371 -1.667134152
## 417 0.80577843 -3.670568261 1.839633609 -1.634515189
## 418 0.80577843 -2.511795523 1.958663532 -1.319198544
## 419 0.80577843 -3.726650277 1.115651608 -1.493166348
## 420 0.80577843 -3.376137680 1.412526239 -1.536658299
## 421 0.80577843 -0.415401588 0.331454470 -0.634200317
## 422 0.80577843 -0.401928760 0.426678408 -0.906025010
## 423 0.80577843 -0.713337295 0.202622083 -0.188407819
## 424 0.80577843 -3.879232792 1.489545601 -0.993008912
## 425 0.80577843 -3.866855315 0.631129805 -1.177849704
## 426 0.80577843 -3.822712635 1.643584324 -1.547531287
## 427 0.80577843 -3.636831424 0.425278056 -1.340944520
## 428 0.80577843 -3.700690438 0.261436868 -1.264833606
## 429 0.80577843 -2.847301799 1.241683291 -1.253960618
## 430 0.80577843 -3.241738006 1.600173411 -1.417055434
## 431 0.80577843 -2.992764527 0.698346703 -0.873406047
## 432 0.80577843 -3.015985986 0.985418869 -0.916897998
## 433 0.80577843 -2.833938506 -0.087250788 -0.699438243
## 434 0.80577843 -2.809402625 0.499496714 -0.895152022
## 435 0.80577843 -2.804583076 0.352459751 -1.177849704
## 436 0.80577843 -2.703591634 1.486744897 -0.993008912
## 437 0.80577843 -3.605723431 0.755761136 -1.406182446
## 438 0.80577843 -3.804748865 1.932056843 -1.504039336
## 439 0.80577843 -3.151590547 2.992123331 -1.536658299
## 440 0.80577843 0.440615895 1.432131167 -1.058246839
## 441 0.80577843 0.380919218 1.324304061 -1.308325557
## 442 0.80577843 0.320784401 0.961612885 -0.590708366
## 443 0.80577843 0.427362137 0.551309739 -0.449359525
## 444 0.80577843 0.329218610 0.867789298 -0.775549157
## 445 0.80577843 -1.272295352 1.559563202 -1.275706593
## 446 0.80577843 -3.435177145 1.586169891 -1.166976716
## 447 0.80577843 -0.423507192 0.719351983 -0.829914096
## 448 0.80577843 0.348825409 0.530304459 -1.079992814
## 449 0.80577843 0.440615895 0.766963952 -0.916897998
## 450 0.80577843 -0.574665749 0.932205492 -1.036500863
## 451 0.80577843 -3.903330533 0.670339662 -0.993008912
## 452 0.80577843 -0.015160016 0.710949871 -0.797295133
## 453 0.80577843 0.311254840 0.646533677 -0.699438243
## 454 0.80577843 0.210263398 0.572315020 -0.514597451
## 455 0.80577843 -3.833666154 0.848184370 -0.829914096
## 456 0.80577843 -3.349082489 0.766963952 -0.916897998
## 457 0.80577843 -3.792042783 0.890194931 -1.069119826
## 458 0.80577843 -3.868498343 0.600322060 -0.982135924
## 459 0.80577843 -0.925178346 0.500897066 -0.829914096
## 460 0.80577843 0.440615895 0.286643205 -0.275391721
## 461 0.80577843 -1.111169093 0.527503755 -0.666819280
## 462 0.80577843 0.380700148 0.279641445 -0.525470439
## 463 0.80577843 0.440615895 0.187218211 -0.329756660
## 464 0.80577843 0.406879058 -0.330912041 -0.253645746
## 465 0.80577843 0.440615895 0.079391104 -0.123169893
## 466 0.80577843 -0.243979021 0.206823139 -0.286264709
## 467 0.80577843 -3.665748713 0.629729453 -0.384121599
## 468 0.80577843 -0.278044464 1.213676250 -0.373248611
## 469 0.80577843 0.132164810 0.766963952 -0.373248611
## 470 0.80577843 0.440615895 0.295045317 -0.264518733
## 471 0.80577843 0.440615895 0.509299178 -0.286264709
## 472 0.80577843 0.423418871 0.030378783 -0.318883672
## 473 0.80577843 0.401949974 0.239031236 0.072543887
## 474 0.80577843 0.197228711 -0.139063813 0.790161078
## 475 0.80577843 -0.044844052 0.768364304 -0.949516961
## 476 0.80577843 -0.590548351 1.602974115 -1.003881900
## 477 0.80577843 0.433057967 0.843983314 -0.634200317
## 478 0.80577843 -0.078799960 1.716402630 -1.145230740
## 479 0.80577843 0.252215374 0.752960432 -0.862533059
## 480 0.80577843 0.291867112 0.063987232 -0.123169893
## 481 0.80577843 0.440615895 -0.267896200 0.050797911
## 482 0.80577843 0.398663919 -0.688001809 0.126908825
## 483 0.80577843 0.422871195 -0.790227508 0.268257666
## 484 0.80577843 0.397020891 -0.312707465 -0.079677942
## 485 0.80577843 0.153962312 0.096195329 -0.210153795
## 486 0.80577843 0.349920761 -0.290301832 -0.144915868
## 487 0.80577843 0.394392046 0.325853062 -0.373248611
## 488 0.80577843 0.345539353 -0.168471205 -0.210153795
## 489 0.75958789 0.420790026 0.757161488 -0.797295133
## 490 0.75958789 -0.138277566 1.584769539 -1.688880128
## 491 0.75958789 -0.418906714 2.384370549 -1.569277262
## 492 0.75958789 0.366241503 0.758561840 -0.971262937
## 493 0.75958789 0.440615895 0.097595681 -0.264518733
## 494 0.34387304 0.440615895 -0.090051492 -0.079677942
## 495 0.34387304 0.440615895 0.131204129 0.213892727
## 496 0.34387304 0.401073693 0.692745294 0.061670899
## 497 0.34387304 0.440615895 1.188469914 -0.308010684
## 498 0.34387304 0.440615895 0.202622083 -0.460232513
## 499 0.34387304 0.440615895 0.037380543 -0.144915868
## 500 0.34387304 0.428238419 0.342657286 -0.547216415
## 501 0.34387304 0.440615895 0.234830180 -0.623327329
## 502 1.17530274 0.386834118 -0.417733867 -0.014440015
## 503 1.17530274 0.440615895 -0.500354637 -0.210153795
## 504 1.17530274 0.440615895 -0.982075736 0.148654801
## 505 1.17530274 0.402826256 -0.864446166 -0.057931966
## 506 1.17530274 0.440615895 -0.668396881 -1.156103728
## attr(,"scaled:center")
## crim zn indus chas nox
## 3.61352356 11.36363636 11.13677866 0.06916996 0.55469506
## rm age dis rad tax
## 6.28463439 68.57490119 3.79504269 9.54940711 408.23715415
## ptratio black lstat medv
## 18.45553360 356.67403162 12.65306324 22.53280632
## attr(,"scaled:scale")
## crim zn indus chas nox rm
## 8.6015451 23.3224530 6.8603529 0.2539940 0.1158777 0.7026171
## age dis rad tax ptratio black
## 28.1488614 2.1057101 8.7072594 168.5371161 2.1649455 91.2948644
## lstat medv
## 7.1410615 9.1971041Next, count the euclidean distance matrix and the summary of distances:
dist_eu <- dist(Boston)
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.119 85.624 170.539 226.315 371.950 626.047K-means clustering algorithm on the distances:
km <-kmeans(Boston, centers = 5)
pairs(Boston, col = km$cluster)
Exploring the optimal number of clusters. First use set.seed() to standardize the k-means protocol:
library (ggplot2)
set.seed(123)
k_max <- 10
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})Draw a plot on the optimal number of clusters:
qplot(x = 1:k_max, y = twcss, geom = 'line')
It seems that 2 clusters would be optimal, since at that point twcss decreases suddenly. Running the algorithm with 2 clusters. Reducing to explore only the last 5 variables
km <-kmeans(Boston, centers = 2)
pairs(Boston[6:10], col=km$cluster)
There is almost a linear correlation between age and distance. ### Bonus LEt’s make sure The Boston data is applicable. Also standardize it by scaling again:
library(MASS)
data("Boston")
scale(Boston)## crim zn indus chas nox
## 1 -0.419366929 0.28454827 -1.28663623 -0.2723291 -0.14407485
## 2 -0.416926670 -0.48724019 -0.59279438 -0.2723291 -0.73953036
## 3 -0.416928995 -0.48724019 -0.59279438 -0.2723291 -0.73953036
## 4 -0.416338404 -0.48724019 -1.30558569 -0.2723291 -0.83445805
## 5 -0.412074053 -0.48724019 -1.30558569 -0.2723291 -0.83445805
## 6 -0.416631374 -0.48724019 -1.30558569 -0.2723291 -0.83445805
## 7 -0.409837246 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 8 -0.403296561 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 9 -0.395543302 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 10 -0.400333140 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 11 -0.393956378 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 12 -0.406444832 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 13 -0.409198989 0.04872402 -0.47618230 -0.2723291 -0.26489191
## 14 -0.346886928 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 15 -0.345933611 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 16 -0.347162460 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 17 -0.297573695 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 18 -0.328932014 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 19 -0.326780075 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 20 -0.335721492 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 21 -0.274570851 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 22 -0.321045059 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 23 -0.276816959 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 24 -0.305188606 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 25 -0.332877817 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 26 -0.322382028 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 27 -0.341986646 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 28 -0.308985598 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 29 -0.330235268 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 30 -0.303558666 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 31 -0.288635766 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 32 -0.262604396 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 33 -0.258736486 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 34 -0.286204807 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 35 -0.232598159 -0.48724019 -0.43682573 -0.2723291 -0.14407485
## 36 -0.412641393 -0.48724019 -0.75459363 -0.2723291 -0.48063666
## 37 -0.408773484 -0.48724019 -0.75459363 -0.2723291 -0.48063666
## 38 -0.410784750 -0.48724019 -0.75459363 -0.2723291 -0.48063666
## 39 -0.399750686 -0.48724019 -0.75459363 -0.2723291 -0.48063666
## 40 -0.416889467 2.72854505 -1.19334657 -0.2723291 -1.09335175
## 41 -0.416196569 2.72854505 -1.19334657 -0.2723291 -1.09335175
## 42 -0.405285738 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 43 -0.403651148 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 44 -0.401574777 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 45 -0.405837965 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 46 -0.400172703 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 47 -0.398203290 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 48 -0.393447167 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 49 -0.390587216 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 50 -0.394551620 -0.48724019 -0.61611679 -0.2723291 -0.92075595
## 51 -0.409786092 0.41317968 -0.80123846 -0.2723291 -0.99842406
## 52 -0.415059564 0.41317968 -0.80123846 -0.2723291 -0.99842406
## 53 -0.413870242 0.41317968 -0.80123846 -0.2723291 -0.99842406
## 54 -0.414310861 0.41317968 -0.80123846 -0.2723291 -0.99842406
## 55 -0.418520570 2.72854505 -1.04029322 -0.2723291 -1.24868797
## 56 -0.418577536 3.37170210 -1.44552019 -0.2723291 -1.30909650
## 57 -0.417712575 3.15731641 -1.51548743 -0.2723291 -1.24868797
## 58 -0.418436864 3.80047346 -1.43094368 -0.2723291 -1.24005818
## 59 -0.402145605 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 60 -0.408094536 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 61 -0.402742009 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 62 -0.400138988 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 63 -0.407281891 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 64 -0.405395021 0.58468822 -0.87557866 -0.2723291 -0.87760700
## 65 -0.417833484 0.26310970 -1.42219777 -0.2723291 -1.19604625
## 66 -0.415934988 2.94293073 -1.13212523 -0.2723291 -1.35224545
## 67 -0.415010735 2.94293073 -1.13212523 -0.2723291 -1.35224545
## 68 -0.413371495 0.04872402 -0.73855947 -0.2723291 -1.25731776
## 69 -0.404344047 0.04872402 -0.73855947 -0.2723291 -1.25731776
## 70 -0.405202032 0.04872402 -0.73855947 -0.2723291 -1.25731776
## 71 -0.409840734 -0.48724019 -0.04763292 -0.2723291 -1.22279860
## 72 -0.401644532 -0.48724019 -0.04763292 -0.2723291 -1.22279860
## 73 -0.409447781 -0.48724019 -0.04763292 -0.2723291 -1.22279860
## 74 -0.397385995 -0.48724019 -0.04763292 -0.2723291 -1.22279860
## 75 -0.410921935 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 76 -0.409043203 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 77 -0.408297988 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 78 -0.409979081 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 79 -0.413537744 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 80 -0.410351107 -0.48724019 0.24681257 -0.2723291 -1.01568364
## 81 -0.415319982 0.58468822 -0.91493524 -0.2723291 -1.11061133
## 82 -0.414914241 0.58468822 -0.91493524 -0.2723291 -1.11061133
## 83 -0.415847794 0.58468822 -0.91493524 -0.2723291 -1.11061133
## 84 -0.415973353 0.58468822 -0.91493524 -0.2723291 -1.11061133
## 85 -0.414220179 -0.48724019 -0.96886832 -0.2723291 -0.91212616
## 86 -0.413434274 -0.48724019 -0.96886832 -0.2723291 -0.91212616
## 87 -0.414070206 -0.48724019 -0.96886832 -0.2723291 -0.91212616
## 88 -0.411788058 -0.48724019 -0.96886832 -0.2723291 -0.91212616
## 89 -0.413521468 -0.48724019 -1.12629463 -0.2723291 -0.56693456
## 90 -0.413937672 -0.48724019 -1.12629463 -0.2723291 -0.56693456
## 91 -0.414656148 -0.48724019 -1.12629463 -0.2723291 -0.56693456
## 92 -0.415530409 -0.48724019 -1.12629463 -0.2723291 -0.56693456
## 93 -0.415215350 0.71331963 0.56895343 -0.2723291 -0.78267931
## 94 -0.416759258 0.71331963 0.56895343 -0.2723291 -0.78267931
## 95 -0.415109555 0.71331963 0.56895343 -0.2723291 -0.78267931
## 96 -0.405913532 -0.48724019 -1.20209248 -0.2723291 -0.94664532
## 97 -0.406727340 -0.48724019 -1.20209248 -0.2723291 -0.94664532
## 98 -0.406054205 -0.48724019 -1.20209248 -0.2723291 -0.94664532
## 99 -0.410583624 -0.48724019 -1.20209248 -0.2723291 -0.94664532
## 100 -0.412126370 -0.48724019 -1.20209248 -0.2723291 -0.94664532
## 101 -0.402818740 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 102 -0.406811046 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 103 -0.393506459 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 104 -0.395500287 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 105 -0.403872039 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 106 -0.404683521 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 107 -0.400198280 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 108 -0.404852095 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 109 -0.405218308 -0.48724019 -0.37560439 -0.2723291 -0.29941107
## 110 -0.389452536 -0.48724019 -0.37560439 -0.2723291 -0.29941107
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## 346 -0.416482564 -0.48724019 -0.98344483 -0.2723291 -0.97253469
## 347 -0.412937852 -0.48724019 -0.98344483 -0.2723291 -0.97253469
## 348 -0.417927653 3.15731641 -1.01842846 -0.2723291 -1.08472196
## 349 -0.418356646 2.94293073 -1.33036576 -0.2723291 -1.03294322
## 350 -0.416731356 1.22784527 -1.44114723 -0.2723291 -1.08472196
## 351 -0.412880885 1.22784527 -1.44114723 -0.2723291 -1.08472196
## 352 -0.410859155 2.08538800 -1.37701059 -0.2723291 -1.24005818
## 353 -0.411679938 2.08538800 -1.37701059 -0.2723291 -1.24005818
## 354 -0.418114829 3.37170210 -1.32890811 -0.2723291 -1.24868797
## 355 -0.415101417 2.94293073 -1.34494227 -0.2723291 -1.22279860
## 356 -0.407709721 2.94293073 -1.34494227 -0.2723291 -1.22279860
## 357 0.624240921 -0.48724019 1.01499462 3.6647712 1.85803641
## 358 0.027457444 -0.48724019 1.01499462 3.6647712 1.85803641
## 359 0.184646645 -0.48724019 1.01499462 3.6647712 1.85803641
## 360 0.075310474 -0.48724019 1.01499462 -0.2723291 1.85803641
## 361 0.107933683 -0.48724019 1.01499462 -0.2723291 1.85803641
## 362 0.025962364 -0.48724019 1.01499462 -0.2723291 1.85803641
## 363 0.007521491 -0.48724019 1.01499462 -0.2723291 1.85803641
## 364 0.070785706 -0.48724019 1.01499462 3.6647712 1.85803641
## 365 -0.016188203 -0.48724019 1.01499462 3.6647712 1.40928733
## 366 0.109555485 -0.48724019 1.01499462 -0.2723291 1.40928733
## 367 0.009699007 -0.48724019 1.01499462 -0.2723291 1.40928733
## 368 1.151964713 -0.48724019 1.01499462 -0.2723291 0.65849561
## 369 0.149356473 -0.48724019 1.01499462 -0.2723291 0.65849561
## 370 0.239079888 -0.48724019 1.01499462 3.6647712 0.65849561
## 371 0.340082672 -0.48724019 1.01499462 3.6647712 0.65849561
## 372 0.653228737 -0.48724019 1.01499462 -0.2723291 0.65849561
## 373 0.541033778 -0.48724019 1.01499462 3.6647712 0.97779784
## 374 0.871305835 -0.48724019 1.01499462 -0.2723291 0.97779784
## 375 1.730465429 -0.48724019 1.01499462 -0.2723291 0.97779784
## 376 1.859616644 -0.48724019 1.01499462 -0.2723291 1.00368721
## 377 1.357253412 -0.48724019 1.01499462 -0.2723291 1.00368721
## 378 0.721959412 -0.48724019 1.01499462 -0.2723291 1.00368721
## 379 2.329195069 -0.48724019 1.01499462 -0.2723291 1.00368721
## 380 1.657048387 -0.48724019 1.01499462 -0.2723291 1.00368721
## 381 9.924109610 -0.48724019 1.01499462 -0.2723291 1.00368721
## 382 1.425427210 -0.48724019 1.01499462 -0.2723291 1.00368721
## 383 0.647964566 -0.48724019 1.01499462 -0.2723291 1.25395112
## 384 0.509089517 -0.48724019 1.01499462 -0.2723291 1.25395112
## 385 1.914932287 -0.48724019 1.01499462 -0.2723291 1.25395112
## 386 1.534407630 -0.48724019 1.01499462 -0.2723291 1.25395112
## 387 2.415877170 -0.48724019 1.01499462 -0.2723291 1.25395112
## 388 2.206996093 -0.48724019 1.01499462 -0.2723291 1.25395112
## 389 1.246308229 -0.48724019 1.01499462 -0.2723291 1.25395112
## 390 0.527604795 -0.48724019 1.01499462 -0.2723291 1.25395112
## 391 0.389305224 -0.48724019 1.01499462 -0.2723291 1.25395112
## 392 0.195258692 -0.48724019 1.01499462 -0.2723291 1.25395112
## 393 0.925923929 -0.48724019 1.01499462 -0.2723291 1.25395112
## 394 0.584922404 -0.48724019 1.01499462 -0.2723291 1.19354259
## 395 1.133084385 -0.48724019 1.01499462 -0.2723291 1.19354259
## 396 0.593291831 -0.48724019 1.01499462 -0.2723291 1.19354259
## 397 0.262572179 -0.48724019 1.01499462 -0.2723291 1.19354259
## 398 0.471833420 -0.48724019 1.01499462 -0.2723291 1.19354259
## 399 4.038608880 -0.48724019 1.01499462 -0.2723291 1.19354259
## 400 0.732778398 -0.48724019 1.01499462 -0.2723291 1.19354259
## 401 2.491712382 -0.48724019 1.01499462 -0.2723291 1.19354259
## 402 1.234973056 -0.48724019 1.01499462 -0.2723291 1.19354259
## 403 0.695478123 -0.48724019 1.01499462 -0.2723291 1.19354259
## 404 2.463298882 -0.48724019 1.01499462 -0.2723291 1.19354259
## 405 4.408007629 -0.48724019 1.01499462 -0.2723291 1.19354259
## 406 7.476247076 -0.48724019 1.01499462 -0.2723291 1.19354259
## 407 1.988326078 -0.48724019 1.01499462 -0.2723291 0.90012973
## 408 0.969311483 -0.48724019 1.01499462 -0.2723291 0.90012973
## 409 0.440661113 -0.48724019 1.01499462 -0.2723291 0.36508275
## 410 1.258468834 -0.48724019 1.01499462 -0.2723291 0.36508275
## 411 5.524853484 -0.48724019 1.01499462 -0.2723291 0.36508275
## 412 1.213407163 -0.48724019 1.01499462 -0.2723291 0.36508275
## 413 1.766830989 -0.48724019 1.01499462 -0.2723291 0.36508275
## 414 2.911369543 -0.48724019 1.01499462 -0.2723291 0.36508275
## 415 4.898256758 -0.48724019 1.01499462 -0.2723291 1.19354259
## 416 1.682381045 -0.48724019 1.01499462 -0.2723291 1.07272553
## 417 0.839462719 -0.48724019 1.01499462 -0.2723291 1.07272553
## 418 2.595705326 -0.48724019 1.01499462 -0.2723291 1.07272553
## 419 8.128839131 -0.48724019 1.01499462 -0.2723291 1.07272553
## 420 0.953174847 -0.48724019 1.01499462 -0.2723291 1.40928733
## 421 0.868899291 -0.48724019 1.01499462 -0.2723291 1.40928733
## 422 0.396331868 -0.48724019 1.01499462 -0.2723291 1.40928733
## 423 0.980600153 -0.48724019 1.01499462 -0.2723291 0.51178918
## 424 0.399567334 -0.48724019 1.01499462 -0.2723291 0.51178918
## 425 0.602054210 -0.48724019 1.01499462 -0.2723291 0.25289548
## 426 1.423787970 -0.48724019 1.01499462 -0.2723291 1.07272553
## 427 1.003735531 -0.48724019 1.01499462 -0.2723291 0.25289548
## 428 3.958402360 -0.48724019 1.01499462 -0.2723291 1.07272553
## 429 0.436385137 -0.48724019 1.01499462 -0.2723291 1.07272553
## 430 0.665620696 -0.48724019 1.01499462 -0.2723291 1.07272553
## 431 0.567177918 -0.48724019 1.01499462 -0.2723291 0.25289548
## 432 0.749723028 -0.48724019 1.01499462 -0.2723291 0.25289548
## 433 0.329071860 -0.48724019 1.01499462 -0.2723291 0.25289548
## 434 0.228743373 -0.48724019 1.01499462 -0.2723291 1.36613839
## 435 1.197444914 -0.48724019 1.01499462 -0.2723291 1.36613839
## 436 0.877386138 -0.48724019 1.01499462 -0.2723291 1.59914271
## 437 1.256434316 -0.48724019 1.01499462 -0.2723291 1.59914271
## 438 1.344372005 -0.48724019 1.01499462 -0.2723291 1.59914271
## 439 1.170089364 -0.48724019 1.01499462 -0.2723291 1.59914271
## 440 0.671635895 -0.48724019 1.01499462 -0.2723291 1.59914271
## 441 2.143519126 -0.48724019 1.01499462 -0.2723291 1.59914271
## 442 0.710413811 -0.48724019 1.01499462 -0.2723291 1.59914271
## 443 0.238660196 -0.48724019 1.01499462 -0.2723291 1.59914271
## 444 0.738590145 -0.48724019 1.01499462 -0.2723291 1.59914271
## 445 1.068270448 -0.48724019 1.01499462 -0.2723291 1.59914271
## 446 0.820582390 -0.48724019 1.01499462 -0.2723291 1.59914271
## 447 0.310937908 -0.48724019 1.01499462 -0.2723291 1.59914271
## 448 0.733743341 -0.48724019 1.01499462 -0.2723291 1.59914271
## 449 0.664481366 -0.48724019 1.01499462 -0.2723291 1.36613839
## 450 0.454858563 -0.48724019 1.01499462 -0.2723291 1.36613839
## 451 0.360888236 -0.48724019 1.01499462 -0.2723291 1.36613839
## 452 0.212475366 -0.48724019 1.01499462 -0.2723291 1.36613839
## 453 0.171672232 -0.48724019 1.01499462 -0.2723291 1.36613839
## 454 0.538806271 -0.48724019 1.01499462 -0.2723291 1.36613839
## 455 0.685935651 -0.48724019 1.01499462 -0.2723291 1.36613839
## 456 0.132400217 -0.48724019 1.01499462 -0.2723291 1.36613839
## 457 0.122688009 -0.48724019 1.01499462 -0.2723291 1.36613839
## 458 0.533282845 -0.48724019 1.01499462 -0.2723291 1.36613839
## 459 0.481158489 -0.48724019 1.01499462 -0.2723291 1.36613839
## 460 0.370589982 -0.48724019 1.01499462 -0.2723291 1.36613839
## 461 0.139347806 -0.48724019 1.01499462 -0.2723291 1.36613839
## 462 0.009252575 -0.48724019 1.01499462 -0.2723291 1.36613839
## 463 0.353587222 -0.48724019 1.01499462 -0.2723291 1.36613839
## 464 0.256654638 -0.48724019 1.01499462 -0.2723291 1.36613839
## 465 0.491283414 -0.48724019 1.01499462 -0.2723291 0.86561057
## 466 -0.052307295 -0.48724019 1.01499462 -0.2723291 0.86561057
## 467 0.018770633 -0.48724019 1.01499462 -0.2723291 0.86561057
## 468 0.094024554 -0.48724019 1.01499462 -0.2723291 0.25289548
## 469 1.390700891 -0.48724019 1.01499462 -0.2723291 0.21837632
## 470 1.099985680 -0.48724019 1.01499462 -0.2723291 0.21837632
## 471 0.085480740 -0.48724019 1.01499462 -0.2723291 0.21837632
## 472 0.049396526 -0.48724019 1.01499462 -0.2723291 -0.19585359
## 473 -0.005213430 -0.48724019 1.01499462 -0.2723291 0.21837632
## 474 0.120137304 -0.48724019 1.01499462 -0.2723291 0.51178918
## 475 0.516449822 -0.48724019 1.01499462 -0.2723291 0.25289548
## 476 0.323150830 -0.48724019 1.01499462 -0.2723291 0.25289548
## 477 0.146239592 -0.48724019 1.01499462 -0.2723291 0.51178918
## 478 1.326491497 -0.48724019 1.01499462 -0.2723291 0.51178918
## 479 0.769568300 -0.48724019 1.01499462 -0.2723291 0.51178918
## 480 1.246308229 -0.48724019 1.01499462 -0.2723291 0.51178918
## 481 0.256987136 -0.48724019 1.01499462 -0.2723291 -0.19585359
## 482 0.243520951 -0.48724019 1.01499462 -0.2723291 -0.19585359
## 483 0.246192564 -0.48724019 1.01499462 -0.2723291 -0.19585359
## 484 -0.092441945 -0.48724019 1.01499462 -0.2723291 -0.19585359
## 485 -0.143573456 -0.48724019 1.01499462 -0.2723291 0.24426569
## 486 0.006992516 -0.48724019 1.01499462 -0.2723291 0.24426569
## 487 0.241610829 -0.48724019 1.01499462 -0.2723291 0.24426569
## 488 0.142084524 -0.48724019 1.01499462 -0.2723291 0.24426569
## 489 -0.402562972 -0.48724019 2.42017014 -0.2723291 0.46864023
## 490 -0.398783418 -0.48724019 2.42017014 -0.2723291 0.46864023
## 491 -0.395982758 -0.48724019 2.42017014 -0.2723291 0.46864023
## 492 -0.407808541 -0.48724019 2.42017014 -0.2723291 0.46864023
## 493 -0.407159820 -0.48724019 2.42017014 -0.2723291 0.46864023
## 494 -0.399952975 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 495 -0.387599381 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 496 -0.399292629 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 497 -0.386433311 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 498 -0.388900310 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 499 -0.392302024 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 500 -0.399427488 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 501 -0.394015670 -0.48724019 -0.21088983 -0.2723291 0.26152527
## 502 -0.412820431 -0.48724019 0.11562398 -0.2723291 0.15796779
## 503 -0.414838673 -0.48724019 0.11562398 -0.2723291 0.15796779
## 504 -0.413037834 -0.48724019 0.11562398 -0.2723291 0.15796779
## 505 -0.407360947 -0.48724019 0.11562398 -0.2723291 0.15796779
## 506 -0.414589880 -0.48724019 0.11562398 -0.2723291 0.15796779
## rm age dis rad tax
## 1 0.413262920 -0.119894767 0.1400749840 -0.9818712 -0.66594918
## 2 0.194082387 0.366803426 0.5566090496 -0.8670245 -0.98635338
## 3 1.281445551 -0.265548971 0.5566090496 -0.8670245 -0.98635338
## 4 1.015297761 -0.809087830 1.0766711351 -0.7521778 -1.10502160
## 5 1.227362043 -0.510674339 1.0766711351 -0.7521778 -1.10502160
## 6 0.206891639 -0.350809969 1.0766711351 -0.7521778 -1.10502160
## 7 -0.388026950 -0.070159185 0.8384142195 -0.5224844 -0.57694801
## 8 -0.160306916 0.977840575 1.0236248974 -0.5224844 -0.57694801
## 9 -0.930285282 1.116389695 1.0861216287 -0.5224844 -0.57694801
## 10 -0.399412952 0.615481336 1.3283202075 -0.5224844 -0.57694801
## 11 0.131459378 0.913894827 1.2117799501 -0.5224844 -0.57694801
## 12 -0.392296701 0.508905089 1.1547920492 -0.5224844 -0.57694801
## 13 -0.563086727 -1.050660656 0.7863652700 -0.5224844 -0.57694801
## 14 -0.477691714 -0.240681180 0.4333252240 -0.6373311 -0.60068166
## 15 -0.268473932 0.565745754 0.3166899868 -0.6373311 -0.60068166
## 16 -0.641365488 -0.428965883 0.3341187865 -0.6373311 -0.60068166
## 17 -0.497617217 -1.395257187 0.3341187865 -0.6373311 -0.60068166
## 18 -0.419338455 0.466274590 0.2198105553 -0.6373311 -0.60068166
## 19 -1.179354069 -1.135921653 0.0006920764 -0.6373311 -0.60068166
## 20 -0.793653261 0.032864520 0.0006920764 -0.6373311 -0.60068166
## 21 -1.017103545 1.048891406 0.0013569352 -0.6373311 -0.60068166
## 22 -0.454919710 0.732715207 0.1031753182 -0.6373311 -0.60068166
## 23 -0.203004422 0.821528746 0.0863638874 -0.6373311 -0.60068166
## 24 -0.671253743 1.116389695 0.1425444597 -0.6373311 -0.60068166
## 25 -0.513272969 0.906789743 0.2871037683 -0.6373311 -0.60068166
## 26 -0.975829289 0.608376252 0.3132232229 -0.6373311 -0.60068166
## 27 -0.671253743 0.771793164 0.4212152950 -0.6373311 -0.60068166
## 28 -0.338213193 0.718505041 0.3126533438 -0.6373311 -0.60068166
## 29 0.299402903 0.917447368 0.3132707128 -0.6373311 -0.60068166
## 30 0.554164692 0.665216917 0.2108349609 -0.6373311 -0.60068166
## 31 -0.813578764 0.906789743 0.2079855659 -0.6373311 -0.60068166
## 32 -0.302631937 1.116389695 0.1804414138 -0.6373311 -0.60068166
## 33 -0.476268464 0.476932215 0.0925850666 -0.6373311 -0.60068166
## 34 -0.830657767 0.938762618 -0.0037244859 -0.6373311 -0.60068166
## 35 -0.268473932 1.006260907 -0.0167367233 -0.6373311 -0.60068166
## 36 -0.500463717 -0.013318520 -0.2064589434 -0.5224844 -0.76681717
## 37 -0.631402737 -0.254891346 -0.1981007179 -0.5224844 -0.76681717
## 38 -0.618593485 -0.961847117 0.0660856927 -0.5224844 -0.76681717
## 39 -0.453496460 -1.363284313 0.0248169544 -0.5224844 -0.76681717
## 40 0.441727925 -1.661697804 0.7627152911 -0.7521778 -0.92701927
## 41 1.052302267 -1.874850298 0.7627152911 -0.7521778 -0.92701927
## 42 0.690796712 -2.333128159 0.9145880470 -0.7521778 -1.03975408
## 43 -0.164576667 -2.201684121 0.9145880470 -0.7521778 -1.03975408
## 44 -0.104800158 -2.205236663 0.9145880470 -0.7521778 -1.03975408
## 45 -0.306901688 -1.015135240 0.9145880470 -0.7521778 -1.03975408
## 46 -0.857699521 -1.235392817 0.6199131095 -0.7521778 -1.03975408
## 47 -0.709681499 -1.253155525 0.6199131095 -0.7521778 -1.03975408
## 48 -0.362408446 0.601271169 0.8996287230 -0.7521778 -1.03975408
## 49 -1.260479332 0.949420242 0.9853955139 -0.7521778 -1.03975408
## 50 -0.971559538 -0.233576097 1.0887810641 -0.7521778 -1.03975408
## 51 -0.457766211 -0.812640371 1.4340327636 -0.6373311 -0.98041997
## 52 -0.241432178 -0.198050682 1.4340327636 -0.6373311 -0.98041997
## 53 0.322174907 -1.686565595 1.4340327636 -0.6373311 -0.98041997
## 54 -0.407952453 -1.675907970 1.4340327636 -0.6373311 -0.98041997
## 55 -0.564509977 -0.745142082 1.6738568465 -0.7521778 0.36053095
## 56 1.372533565 -1.658145262 2.3277455193 -0.5224844 -1.08128796
## 57 0.139998879 -1.167894527 2.5609210138 -0.8670245 -0.56508119
## 58 0.756266222 -0.997372532 2.1511780064 -0.5224844 -0.90328562
## 59 -0.198734672 -1.398809729 1.9089794276 -0.1779443 -0.73715011
## 60 -0.509003218 -0.759352248 1.4897384367 -0.1779443 -0.73715011
## 61 -0.773727758 -0.084369352 1.6290738544 -0.1779443 -0.73715011
## 62 -0.453496460 0.881921953 1.4358373805 -0.1779443 -0.73715011
## 63 0.243896145 -0.027528687 1.6291213443 -0.1779443 -0.73715011
## 64 0.679410711 -0.894348827 1.9878601804 -0.1779443 -0.73715011
## 65 1.166162284 -0.322389636 2.5776849546 -0.7521778 -1.14062207
## 66 0.007636609 -1.803799466 1.3375332514 -0.6373311 -0.42267932
## 67 -0.708258248 -1.331311439 1.3375332514 -0.6373311 -0.42267932
## 68 -0.578742479 -1.675907970 1.2836321952 -0.6373311 -0.37521203
## 69 -0.982945540 -1.128816570 1.2836321952 -0.6373311 -0.37521203
## 70 -0.568779727 -1.263813149 1.2836321952 -0.6373311 -0.37521203
## 71 0.188389387 -2.201684121 0.7086717651 -0.6373311 -0.61254848
## 72 -0.460612711 -1.814457091 0.7086717651 -0.6373311 -0.61254848
## 73 -0.312594689 -2.159053622 0.7086717651 -0.6373311 -0.61254848
## 74 -0.056409650 -2.215894287 0.7086717651 -0.6373311 -0.61254848
## 75 -0.016558644 -2.222999370 0.2167712006 -0.5224844 -0.06074124
## 76 0.001943608 -0.837508162 0.3360183832 -0.5224844 -0.06074124
## 77 -0.008019143 0.210491598 0.1221237952 -0.5224844 -0.06074124
## 78 -0.205850923 -0.809087830 0.1403124336 -0.5224844 -0.06074124
## 79 -0.074911903 -0.528437047 0.5789293108 -0.5224844 -0.06074124
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## 305 1.354031312 -0.976057283 0.1077818401 -0.2927910 -1.10502160
## 306 0.471616179 -0.372125218 -0.2018524214 -0.2927910 -1.10502160
## 307 1.615909352 0.118125517 -0.3304551177 -0.2927910 -1.10502160
## 308 0.803233479 0.061284852 -0.2908010367 -0.2927910 -1.10502160
## 309 0.498657933 0.494694923 -0.2267846280 -0.6373311 -0.61848189
## 310 -0.444956959 0.288647512 -0.3288879504 -0.6373311 -0.61848189
## 311 -1.866783923 -1.093291155 -0.6058016588 -0.6373311 -0.61848189
## 312 -0.231469427 -0.560409921 -0.5483863487 -0.6373311 -0.61848189
## 313 -0.372371198 0.775345706 -0.4563983787 -0.6373311 -0.61848189
## 314 -0.026521396 0.505352547 -0.2527616128 -0.6373311 -0.61848189
## 315 0.401876919 0.665216917 -0.0915333432 -0.6373311 -0.61848189
## 316 -0.824964766 0.324172928 0.0712146037 -0.6373311 -0.61848189
## 317 -0.527505471 0.519562713 0.0966691995 -0.6373311 -0.61848189
## 318 -0.715374500 0.111020434 0.1123883621 -0.6373311 -0.61848189
## 319 0.138575629 -0.048843936 -0.1246813056 -0.6373311 -0.61848189
## 320 -0.244278679 -0.347257427 0.0982363667 -0.6373311 -0.61848189
## 321 0.201198639 -0.578172628 0.3539695720 -0.5224844 -0.71934988
## 322 0.130036128 -0.507121797 0.3539695720 -0.5224844 -0.71934988
## 323 -0.346752694 -0.663433626 0.4397838527 -0.5224844 -0.71934988
## 324 -0.820695015 0.203386515 0.4397838527 -0.5224844 -0.71934988
## 325 0.185542886 -1.011582699 0.4397838527 -0.5224844 -0.71934988
## 326 0.208314890 -1.913928255 0.7697437989 -0.5224844 -0.71934988
## 327 0.038948114 -1.409467353 0.7697437989 -0.5224844 -0.71934988
## 328 -0.286976185 -0.883691203 0.7697437989 -0.5224844 -0.71934988
## 329 -0.592974981 -1.519596142 0.6741465952 -0.6373311 0.12912791
## 330 0.068836369 -1.825114716 0.6741465952 -0.6373311 0.12912791
## 331 -0.200157922 -1.292233482 0.9871051509 -0.6373311 0.12912791
## 332 -0.823541516 -1.427230061 1.3514003073 -0.9818712 -0.61848189
## 333 -0.360985196 -1.608409681 1.3514003073 -0.9818712 -0.61848189
## 334 0.044641115 -1.082633530 1.2648261879 -0.5224844 -1.09315478
## 335 0.036101614 -1.068423364 1.2648261879 -0.5224844 -1.09315478
## 336 -0.352445695 -1.210525026 1.0401513886 -0.5224844 -1.09315478
## 337 -0.591551731 -0.791325122 0.6819824315 -0.5224844 -1.09315478
## 338 -0.554547225 -0.318837095 0.8642962245 -0.5224844 -1.09315478
## 339 -0.321134190 -1.111053862 0.4830471675 -0.5224844 -1.09315478
## 340 -0.426454706 -0.823297996 0.4830471675 -0.5224844 -1.09315478
## 341 -0.450649960 -0.357915052 0.4830471675 -0.5224844 -1.09315478
## 342 1.361147563 -0.684748875 1.5400302593 -0.9818712 -0.73715011
## 343 0.363449163 -0.315284553 1.1738829960 -0.9818712 0.08166062
## 344 0.585476196 -0.432518424 0.9199069177 -0.5224844 -0.22687676
## 345 0.838814735 -1.437887686 1.2681504821 -0.5224844 -0.22687676
## 346 -0.385180450 -0.713169208 2.0033893834 -0.7521778 -0.33367816
## 347 -0.550277475 -0.578172628 2.0033893834 -0.7521778 -0.33367816
## 348 0.329291158 -1.452097852 2.2511442825 -0.6373311 -0.33961157
## 349 0.498657933 -1.381047021 2.1602960705 -0.6373311 -0.76088376
## 350 0.931325998 -1.210525026 2.3730983904 -0.9818712 -0.43454615
## 351 0.292286652 -0.858823412 2.3730983904 -0.9818712 -0.43454615
## 352 0.418955921 -1.160789444 3.2840499862 -0.6373311 0.01639310
## 353 -0.570202978 -1.778931675 3.2840499862 -0.6373311 0.01639310
## 354 0.631020203 -1.153684361 3.9566021965 -0.5224844 -1.31269099
## 355 -0.884741275 -1.658145262 3.2248775491 -0.6373311 -0.44047956
## 356 -0.496193967 -1.743406260 3.2248775491 -0.6373311 -0.44047956
## 357 -0.103376907 1.024023615 -0.7944316108 1.6596029 1.52941294
## 358 0.157077882 0.796660955 -0.6125452271 1.6596029 1.52941294
## 359 -0.224353176 0.526667797 -0.5092546567 1.6596029 1.52941294
## 360 -0.245701929 0.452064424 -0.6106931203 1.6596029 1.52941294
## 361 0.161347633 0.690084708 -0.6063715378 1.6596029 1.52941294
## 362 -0.047870149 0.800213497 -0.7121315839 1.6596029 1.52941294
## 363 -1.313139590 0.981393116 -0.8032647354 1.6596029 1.52941294
## 364 -0.685486245 0.725610124 -0.8977221812 1.6596029 1.52941294
## 365 3.551529643 0.508905089 -0.8977221812 1.6596029 1.52941294
## 366 -3.876413226 0.686532167 -1.0361552904 1.6596029 1.52941294
## 367 -1.881016425 0.810871121 -0.9700968153 1.6596029 1.52941294
## 368 -3.446591661 1.116389695 -1.0848799457 1.6596029 1.52941294
## 369 -1.871053674 1.116389695 -1.1694594886 1.6596029 1.52941294
## 370 0.566973944 1.002708366 -1.1579669285 1.6596029 1.52941294
## 371 1.040916265 1.027576156 -1.2312438711 1.6596029 1.52941294
## 372 -0.097683907 1.116389695 -1.2470580136 1.6596029 1.52941294
## 373 -0.583012230 0.746925373 -1.2658165310 1.6596029 1.52941294
## 374 -1.962141687 1.116389695 -1.2446360278 1.6596029 1.52941294
## 375 -3.055197852 1.116389695 -1.2623022771 1.6596029 1.52941294
## 376 1.463621579 1.041786323 -1.1771528552 1.6596029 1.52941294
## 377 0.518583436 0.878369411 -1.1635707388 1.6596029 1.52941294
## 378 0.724954717 1.073759197 -1.1573495596 1.6596029 1.52941294
## 379 0.135729129 0.981393116 -1.1440048928 1.6596029 1.52941294
## 380 -0.087721155 1.116389695 -1.1440048928 1.6596029 1.52941294
## 381 0.972600255 0.828633829 -1.1295679579 1.6596029 1.52941294
## 382 0.370565414 1.084416821 -1.0807958128 1.6596029 1.52941294
## 383 -1.065494052 1.116389695 -1.0517319833 1.6596029 1.52941294
## 384 -1.088266056 1.116389695 -1.0741947142 1.6596029 1.52941294
## 385 -2.727850303 0.803766038 -1.1186452769 1.6596029 1.52941294
## 386 -1.434115858 1.048891406 -1.1250089259 1.6596029 1.52941294
## 387 -2.323647242 1.116389695 -1.1054905698 1.6596029 1.52941294
## 388 -1.828356167 0.743372832 -1.0811757321 1.6596029 1.52941294
## 389 -1.999146193 1.116389695 -1.0474104008 1.6596029 1.52941294
## 390 -1.273288584 1.077311738 -0.9815893753 1.6596029 1.52941294
## 391 -0.813578764 1.009813449 -0.8873693792 1.6596029 1.52941294
## 392 -0.332520192 0.494694923 -0.7727762084 1.6596029 1.52941294
## 393 -1.777119159 1.009813449 -0.9616910999 1.6596029 1.52941294
## 394 -0.130418661 0.853501620 -0.9516232374 1.6596029 1.52941294
## 395 -0.565933227 0.928104993 -0.9559448199 1.6596029 1.52941294
## 396 0.265244898 1.073759197 -0.9827291333 1.6596029 1.52941294
## 397 0.171310384 0.974288033 -1.0059517029 1.6596029 1.52941294
## 398 -0.765188257 1.077311738 -1.0265623271 1.6596029 1.52941294
## 399 -1.183623820 1.116389695 -1.0948528283 1.6596029 1.52941294
## 400 -0.615746985 0.327725469 -1.0897239172 1.6596029 1.52941294
## 401 -0.423608206 1.116389695 -1.0477428302 1.6596029 1.52941294
## 402 0.083068871 1.116389695 -1.0547238481 1.6596029 1.52941294
## 403 0.169887134 1.116389695 -1.0239028917 1.6596029 1.52941294
## 404 -1.331641842 0.974288033 -0.9936043244 1.6596029 1.52941294
## 405 -1.072610303 0.597718628 -1.0389097056 1.6596029 1.52941294
## 406 -0.856276271 1.116389695 -1.1253413553 1.6596029 1.52941294
## 407 -3.055197852 1.116389695 -1.2427839210 1.6596029 1.52941294
## 408 -0.963020037 1.116389695 -1.1919222195 1.6596029 1.52941294
## 409 -0.950210785 1.041786323 -1.1114268095 1.6596029 1.52941294
## 410 0.807503230 1.116389695 -1.1062978984 1.6596029 1.52941294
## 411 -0.750955755 1.116389695 -1.1312301050 1.6596029 1.52941294
## 412 0.529969438 1.116389695 -1.0768541496 1.6596029 1.52941294
## 413 -2.357805247 1.116389695 -1.0643168114 1.6596029 1.52941294
## 414 -1.607752384 1.116389695 -1.0474578907 1.6596029 1.52941294
## 415 -2.512939520 1.116389695 -1.0147848276 1.6596029 1.52941294
## 416 0.212584640 1.116389695 -0.9309651233 1.6596029 1.52941294
## 417 0.707875715 0.789555872 -0.9381835908 1.6596029 1.52941294
## 418 -1.395688102 0.729162665 -1.0198662487 1.6596029 1.52941294
## 419 -0.466305712 1.116389695 -0.9462093868 1.6596029 1.52941294
## 420 0.767652224 0.281542429 -0.9502935197 1.6596029 1.52941294
## 421 0.179849885 1.116389695 -0.9194725633 1.6596029 1.52941294
## 422 -0.396566452 0.949420242 -0.9120166463 1.6596029 1.52941294
## 423 -0.906090028 0.675874542 -0.8756393696 1.6596029 1.52941294
## 424 -0.258511181 0.587061003 -0.8421114879 1.6596029 1.52941294
## 425 -1.024219796 0.071942477 -0.8223081923 1.6596029 1.52941294
## 426 -0.553123975 0.952972784 -0.8953951752 1.6596029 1.52941294
## 427 -0.637095738 -0.315284553 -0.8536040479 1.6596029 1.52941294
## 428 -0.117609410 0.359698343 -0.9175729666 1.6596029 1.52941294
## 429 -0.130418661 0.338383094 -0.8830477967 1.6596029 1.52941294
## 430 0.135729129 0.960077867 -0.8675660836 1.6596029 1.52941294
## 431 0.090185122 0.622586419 -0.8274371034 1.6596029 1.52941294
## 432 0.780461476 0.913894827 -0.8105781827 1.6596029 1.52941294
## 433 0.199775388 0.221149222 -0.7572944954 1.6596029 1.52941294
## 434 0.215431141 0.686532167 -0.7024911307 1.6596029 1.52941294
## 435 -0.109069908 0.938762618 -0.7469416934 1.6596029 1.52941294
## 436 0.490118432 0.924552451 -0.7932443629 1.6596029 1.52941294
## 437 0.251012396 0.878369411 -0.8512295520 1.6596029 1.52941294
## 438 -0.188771920 1.116389695 -0.8932106390 1.6596029 1.52941294
## 439 -0.497617217 0.686532167 -0.9376612017 1.6596029 1.52941294
## 440 -0.935978283 0.899684660 -0.9392758589 1.6596029 1.52941294
## 441 -0.664137492 0.846396537 -0.9160057994 1.6596029 1.52941294
## 442 0.172733634 1.016918532 -0.8215483536 1.6596029 1.52941294
## 443 -0.093414156 1.116389695 -0.8501847738 1.6596029 1.52941294
## 444 0.285170401 1.116389695 -0.8627221120 1.6596029 1.52941294
## 445 -0.612900484 0.995603282 -0.9020437636 1.6596029 1.52941294
## 446 0.248165896 0.931657534 -0.8582105699 1.6596029 1.52941294
## 447 0.080222370 0.988498199 -0.8182715493 1.6596029 1.52941294
## 448 -0.047870149 0.995603282 -0.7584342534 1.6596029 1.52941294
## 449 -0.141804663 1.070206655 -0.7282306659 1.6596029 1.52941294
## 450 0.188389387 1.055996489 -0.7646079427 1.6596029 1.52941294
## 451 0.660908458 0.853501620 -0.6987869171 1.6596029 1.52941294
## 452 0.527122938 1.052443947 -0.6837801032 1.6596029 1.52941294
## 453 0.017599361 0.825081288 -0.6776064140 1.6596029 1.52941294
## 454 1.577481596 1.091521905 -0.6374774338 1.6596029 1.52941294
## 455 0.631020203 0.906789743 -0.6168668096 1.6596029 1.52941294
## 456 0.342100410 0.636796585 -0.6455032298 1.6596029 1.52941294
## 457 -0.439263958 0.686532167 -0.5767378294 1.6596029 1.52941294
## 458 -0.496193967 0.416539008 -0.4824228534 1.6596029 1.52941294
## 459 0.023292362 0.537325421 -0.4805707466 1.6596029 1.52941294
## 460 -0.289822685 0.562193212 -0.5117241325 1.6596029 1.52941294
## 461 0.592592447 0.761135540 -0.5687120333 1.6596029 1.52941294
## 462 0.130036128 0.704294875 -0.5831489682 1.6596029 1.52941294
## 463 0.046064365 0.512457630 -0.5036983364 1.6596029 1.52941294
## 464 0.325021407 0.757582998 -0.4717851119 1.6596029 1.52941294
## 465 -0.107646658 -0.112789684 -0.3949464255 1.6596029 1.52941294
## 466 -0.748109254 -0.723826832 -0.3459843207 1.6596029 1.52941294
## 467 -0.473421963 0.572850837 -0.4385896596 1.6596029 1.52941294
## 468 -0.400836202 0.920999910 -0.5958762661 1.6596029 1.52941294
## 469 -0.510426469 0.086152643 -0.4210658801 1.6596029 1.52941294
## 470 -0.813578764 -0.421860800 -0.4612898402 1.6596029 1.52941294
## 471 -0.167423167 0.547983046 -0.3617034834 1.6596029 1.52941294
## 472 -0.079181654 0.786003330 -0.3304076278 1.6596029 1.52941294
## 473 0.216854391 0.228254306 -0.4267171803 1.6596029 1.52941294
## 474 0.989679257 -0.034633770 -0.5993905200 1.6596029 1.52941294
## 475 -1.220628326 0.952972784 -0.6483526248 1.6596029 1.52941294
## 476 -0.174539418 1.024023615 -0.7546350600 1.6596029 1.52941294
## 477 0.283747151 0.889027036 -0.7074775720 1.6596029 1.52941294
## 478 -1.395688102 1.020471073 -0.8046419430 1.6596029 1.52941294
## 479 -0.141804663 0.999155824 -0.7714939807 1.6596029 1.52941294
## 480 -0.079181654 0.690084708 -0.8756393696 1.6596029 1.52941294
## 481 -0.060679401 -0.137657475 -0.1761128861 1.6596029 1.52941294
## 482 0.662331708 0.224701764 -0.2200410597 1.6596029 1.52941294
## 483 1.104962525 0.299305137 -0.1825715149 1.6596029 1.52941294
## 484 -0.743839504 -1.004477616 0.1440166471 1.6596029 1.52941294
## 485 -0.588705230 -0.947636951 -0.0337381137 1.6596029 1.52941294
## 486 0.038948114 -0.592382795 0.0933923952 1.6596029 1.52941294
## 487 -0.242855428 0.398776300 -0.1183176566 1.6596029 1.52941294
## 488 -0.540314723 -0.546199754 -0.3052379716 1.6596029 1.52941294
## 489 -1.182200570 0.857054162 -0.9375187319 -0.6373311 1.79641644
## 490 -1.239130578 1.055996489 -0.9686246278 -0.6373311 1.79641644
## 491 -1.695993897 1.045338864 -0.9367114033 -0.6373311 1.79641644
## 492 -0.429301206 1.073759197 -0.9151034909 -0.6373311 1.79641644
## 493 -0.429301206 0.530220338 -0.8002728706 -0.6373311 1.79641644
## 494 -0.822118266 -0.517779422 -0.6711952751 -0.4076377 -0.10227512
## 495 -0.510426469 -0.922769160 -0.6711952751 -0.4076377 -0.10227512
## 496 -0.874778524 -1.413019895 -0.4732098094 -0.4076377 -0.10227512
## 497 -1.273288584 0.153650933 -0.4732098094 -0.4076377 -0.10227512
## 498 -0.698295497 0.071942477 -0.4285217972 -0.4076377 -0.10227512
## 499 -0.378064199 -0.116342226 -0.6581830377 -0.4076377 -0.10227512
## 500 -1.018526795 0.174966182 -0.6625521101 -0.4076377 -0.10227512
## 501 -0.366678197 0.395223759 -0.6158695213 -0.4076377 -0.10227512
## 502 0.438881424 0.018654354 -0.6251775451 -0.9818712 -0.80241764
## 503 -0.234315927 0.288647512 -0.7159307773 -0.9818712 -0.80241764
## 504 0.983986256 0.796660955 -0.7729186782 -0.9818712 -0.80241764
## 505 0.724954717 0.736267749 -0.6677760011 -0.9818712 -0.80241764
## 506 -0.362408446 0.434301716 -0.6126402069 -0.9818712 -0.80241764
## ptratio black lstat medv
## 1 -1.45755797 0.440615895 -1.074498970 0.159527789
## 2 -0.30279450 0.440615895 -0.491952525 -0.101423917
## 3 -0.30279450 0.396035074 -1.207532413 1.322937477
## 4 0.11292035 0.415751408 -1.360170785 1.181588636
## 5 0.11292035 0.440615895 -1.025486649 1.486032293
## 6 0.11292035 0.410165113 -1.042290874 0.670558212
## 7 -1.50374851 0.426376321 -0.031236706 0.039924923
## 8 -1.50374851 0.440615895 0.909799859 0.496590409
## 9 -1.50374851 0.328123258 2.419379350 -0.655946292
## 10 -1.50374851 0.328999540 0.622727693 -0.394994586
## 11 -1.50374851 0.392639483 1.091845624 -0.819041108
## 12 -1.50374851 0.440615895 0.086392864 -0.394994586
## 13 -1.50374851 0.370513376 0.428078760 -0.090550930
## 14 1.17530274 0.440615895 -0.615183504 -0.231899770
## 15 1.17530274 0.255720500 -0.335113097 -0.471105500
## 16 1.17530274 0.426595391 -0.585776111 -0.286264709
## 17 1.17530274 0.330533033 -0.850442645 0.061670899
## 18 1.17530274 0.329437681 0.282442149 -0.547216415
## 19 1.17530274 -0.741378303 -0.134862757 -0.253645746
## 20 1.17530274 0.375442459 -0.192277190 -0.471105500
## 21 1.17530274 0.217930861 1.171665689 -0.971262937
## 22 1.17530274 0.392749018 0.164812578 -0.318883672
## 23 1.17530274 0.440615895 0.849584722 -0.797295133
## 24 1.17530274 0.414765591 1.012025558 -0.873406047
## 25 1.17530274 0.412465352 0.510699530 -0.753803182
## 26 1.17530274 -0.583319029 0.540106923 -0.938643973
## 27 1.17530274 0.221326452 0.302047077 -0.645073304
## 28 1.17530274 -0.550896614 0.647934029 -0.840787084
## 29 1.17530274 0.342472368 0.020576319 -0.449359525
## 30 1.17530274 0.258020739 -0.094252548 -0.166661844
## 31 1.17530274 0.038293155 1.392921311 -1.069119826
## 32 1.17530274 0.219683424 0.054184768 -0.873406047
## 33 1.17530274 -1.359047220 2.108501199 -1.014754888
## 34 1.17530274 0.022958229 0.797771697 -1.025627875
## 35 1.17530274 -1.186967442 1.076441751 -0.982135924
## 36 0.34387304 0.440615895 -0.416333515 -0.394994586
## 37 0.34387304 0.228774844 -0.174072614 -0.275391721
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## 506 1.17530274 0.440615895 -0.668396881 -1.156103728
## attr(,"scaled:center")
## crim zn indus chas nox
## 3.61352356 11.36363636 11.13677866 0.06916996 0.55469506
## rm age dis rad tax
## 6.28463439 68.57490119 3.79504269 9.54940711 408.23715415
## ptratio black lstat medv
## 18.45553360 356.67403162 12.65306324 22.53280632
## attr(,"scaled:scale")
## crim zn indus chas nox rm
## 8.6015451 23.3224530 6.8603529 0.2539940 0.1158777 0.7026171
## age dis rad tax ptratio black
## 28.1488614 2.1057101 8.7072594 168.5371161 2.1649455 91.2948644
## lstat medv
## 7.1410615 9.1971041Next let’s make k clustering on the scaled Boston data with 3 centers:
km3 <-kmeans(Boston, centers = 3)
km3## K-means clustering with 3 clusters of sizes 268, 101, 137
##
## Cluster means:
## crim zn indus chas nox rm age
## 1 0.2410479 17.817164 6.668582 0.07462687 0.4833981 6.465448 55.70522
## 2 0.7807617 9.653465 13.070594 0.06930693 0.5873366 6.182515 73.70594
## 3 12.2991617 0.000000 18.451825 0.05839416 0.6701022 6.006212 89.96788
## dis rad tax ptratio black lstat medv
## 1 4.873560 4.313433 276.5485 17.87313 387.8141 9.538022 25.86530
## 2 3.294209 4.831683 405.8020 17.63960 363.0747 12.750990 22.18218
## 3 2.054470 23.270073 667.6423 20.19635 291.0391 18.674526 16.27226
##
## Clustering vector:
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
## 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
## 1 1 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1
## 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
## 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
## 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
## 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1
## 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
## 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
## 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1
## 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198
## 1 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1
## 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
## 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306
## 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1
## 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342
## 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1
## 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
## 2 2 2 2 2 1 1 1 1 2 2 1 1 1 3 3 3 3
## 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504
## 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 1
## 505 506
## 1 1
##
## Within cluster sum of squares by cluster:
## [1] 924118.8 640601.1 2896224.4
## (between_SS / total_SS = 77.0 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss"
## [5] "tot.withinss" "betweenss" "size" "iter"
## [9] "ifault"An arrow plot on the linear discriminant analysis
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "orange", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}Superbonus
Let’s create a matrix of model predictors on crime in “train” data set:
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutmodel_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)Next, we’ll adjust the code to include crime classes in the plot
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', col="classes")## Warning: 'scatter3d' objects don't have these attributes: 'col'
## Valid attributes include:
## 'type', 'visible', 'showlegend', 'legendgroup', 'opacity', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'transforms', 'uirevision', 'x', 'y', 'z', 'text', 'hovertext', 'hovertemplate', 'mode', 'surfaceaxis', 'surfacecolor', 'projection', 'connectgaps', 'line', 'marker', 'textposition', 'textfont', 'hoverinfo', 'error_x', 'error_y', 'error_z', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'xsrc', 'ysrc', 'zsrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'textpositionsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', col="clusters")## Warning: 'scatter3d' objects don't have these attributes: 'col'
## Valid attributes include:
## 'type', 'visible', 'showlegend', 'legendgroup', 'opacity', 'name', 'uid', 'ids', 'customdata', 'meta', 'hoverlabel', 'stream', 'transforms', 'uirevision', 'x', 'y', 'z', 'text', 'hovertext', 'hovertemplate', 'mode', 'surfaceaxis', 'surfacecolor', 'projection', 'connectgaps', 'line', 'marker', 'textposition', 'textfont', 'hoverinfo', 'error_x', 'error_y', 'error_z', 'xcalendar', 'ycalendar', 'zcalendar', 'scene', 'idssrc', 'customdatasrc', 'metasrc', 'xsrc', 'ysrc', 'zsrc', 'textsrc', 'hovertextsrc', 'hovertemplatesrc', 'textpositionsrc', 'hoverinfosrc', 'key', 'set', 'frame', 'transforms', '_isNestedKey', '_isSimpleKey', '_isGraticule', '_bbox'Dimension reduction
To start with dimension reduction, first let’s read the data:
human<-read.csv("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/human2.txt", header=TRUE, sep= ",")
str(human)## 'data.frame': 155 obs. of 8 variables:
## $ Edu2.FM : num 1.007 0.997 0.983 0.989 0.969 ...
## $ Labo.FM : num 0.891 0.819 0.825 0.884 0.829 ...
## $ Edu.Exp : num 17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
## $ Life.Exp : num 81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
## $ GNI : int 64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
## $ Mat.Mor : int 4 6 6 5 6 7 9 28 11 8 ...
## $ Ado.Birth: num 7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
## $ Parli.F : num 39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...Item 1 Visualization of the data.
library(GGally)
library(dplyr)Let us calculate a correlation matrix and then using the %>% -function draw a correlation plot:
library(corrplot)
cor(human) %>%corrplot()
Here we can see that Life expectancy at birth and expected years of schooling have a strong positive correlation. Maternal mortality rate and adolescent birth rate also have a strong positive correlation. Maternal mortality has strongest negative correlation with life expectancy, and also a quite strong negative correlation with female/male proportion of secondary education, and expected years of schooling. Adolescent birth rate and life expectancy correlate negatively.
Item 2. Principal component analysis on unstandardized data “human”
pca_human <- prcomp(human)
biplot(pca_human, choices = 1:2)## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
s <- summary(pca_human)
pca_pr <- round(100*s$importance[2, ], digits = 1)R can’t properly produce the PCA on unstandardized data. Here the arrow just shows strong negative correlation between GNI and prinicipal component 1, the other variables remaining unclear. So let’s standardize the data:
Item 3. Working with standardized data and biplotting it:
human_std <- scale(human)
pca_human <- prcomp(human_std)
biplot(pca_human, choices = 1:2)
The results seem a lot different after standardization. The reason is that PCA is sensitive to scaling of original features, and assumes that variables with more variance would be more significant than other. Thus, standardization of hte data is important.
In biplot, angle between variables shows the degree of correlation. Angle betwwn variable and principal component axes also similarly show correlation. Small angle= high correlation. Here we can see that maternal mortality and adolescent birth rate have high positive correlation to each other, and also to the PC1. Expected time in education, life expectancy and female/male education ratio have a positive correlation to each other, negative correalation with adolescence birth and maternal mortality ratio, as well as a negative correlation with PC1.
The female/male ratio of labour force participation and ratio of females in parliament show high positive correlation both with each other and PC2.
Item 4 interpretation of the first two principal components
It seems that principal component 1 (PC1) is related with (un)survival, as it correlates positively with such societal issues as maternal mortality rate or adolescents births. Thus it has a negative correlation with issues that are related with better survival, such as life expectancy and expected years of schooling.
Principal component 2 (PC2) is related with variables that show higher levels of Maslow’s need hierarchy, such as participation in work froce (described by female/male ratio) or female participation in parliament. The variables related to PC2 thus show some higher developmental stage of the society.
Item 5
Read data “Teatime” from the already installed FactoMineR -library:
library("FactoMineR")data("tea")str(tea)## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
## $ frequency : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...dim(tea)## [1] 300 36library(tidyr)
library(dplyr)
library(ggplot2)gather(tea) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar()## Warning: attributes are not identical across measure variables;
## they will be dropped
Since the data is large, and difficult to read, let’s reduce the number of columns, and keep the following:
keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")LEt’s create a data set “tea_time”
tea_time <- select(tea,one_of(keep_columns))str(tea_time)## 'data.frame': 300 obs. of 6 variables:
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ sugar: Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ where: Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ lunch: Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...summary(tea_time)## Tea How how sugar
## black : 74 alone:195 tea bag :170 No.sugar:155
## Earl Grey:193 lemon: 33 tea bag+unpackaged: 94 sugar :145
## green : 33 milk : 63 unpackaged : 36
## other: 9
## where lunch
## chain store :192 lunch : 44
## chain store+tea shop: 78 Not.lunch:256
## tea shop : 30
## Now see how this looks as a plot:
gather(tea_time) %>% ggplot(aes(value), theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 4))) + facet_wrap("key", scales = "free") + geom_bar() ## Warning: attributes are not identical across measure variables;
## they will be dropped
The bars show distribution of consumers different habits related with tea consumption: Most common is to prepare it from tea bag, drink it alone, not related with lunch. Sugar vs no sugar are almost even, whereas Earl Grey is the most popular quality, and tea is mostly bought from chain stores.
To perform a multiple correspondence analysis on the “tea_time” data, and see the summary of the model:
mca <- MCA(tea_time, graph = FALSE)
summary(mca)##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.279 0.261 0.219 0.189 0.177 0.156
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953
## Dim.7 Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.144 0.141 0.117 0.087 0.062
## % of var. 7.841 7.705 6.392 4.724 3.385
## Cumulative % of var. 77.794 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898
## cos2 v.test Dim.3 ctr cos2 v.test
## black 0.003 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 0.027 2.867 | 0.433 9.160 0.338 10.053 |
## green 0.107 -5.669 | -0.108 0.098 0.001 -0.659 |
## alone 0.127 -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 0.035 3.226 | 1.329 14.771 0.218 8.081 |
## milk 0.020 2.422 | 0.013 0.003 0.000 0.116 |
## other 0.102 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag 0.161 -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 0.478 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged 0.141 -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |Of the categories, based on the coordinates of categories, earl grey, alone, tea bag, and milk are located close to each other, on the negative coordinate. From v-test results we can see that dimension 1 has a strong negative correlation with teabags, and strong positive correlation with unpackaged tea. Earl grey is also negatively correlated with Dim 1, whereas Black tea correlates positively.
Draw a plot of the MCA, in which the variables are shown in different colors:
plot(mca, invisible=c("ind"), habillage="quali")
The plot shows similar results than the written output: Similar coordinates (correlation) for Earl Grey and milk; for sugar, chain store and tea bag. In the right lower corner, tea shop is located together with green, unpackaged, and alone. These correlation seem to predict cunsumer behavior: the way they drink their tea, its packaging style, and the place where they but it from correlate with each other.